Before we start on on the Assault in specific: soldier stuff!
First of all, all your soldiers start as pathetic sadsack Rookies. Much like classic X-COM, the notion that you're an elite paramilitary organization pulling from the best across the world exists in the lore but is poorly-supported by gameplay: your Rookies are all barely competent at using their Rifle and Pistol and have absolutely no abilities beyond the standard mechanics the game passes out to most any human-ish Cover-using unit. (Such as Hunker Down, reloading their primary weapon, entering Overwatch...) They need to get some kills in before they'll get a chance to be even slightly good at stuff.
Technically speaking, soldiers don't strictly need to get kills to level up. Just completing a mission at all gives 60 experience to each soldier, or even 90 if they killed an 'elite' alien during the mission while being below Sergeant rank, and they additionally get 20 experience if no XCOM soldier died during the mission. Certain storylines missions give an even bigger reward, of 120 experience or 180 if they killed an 'elite' alien while below Sergeant rank.
However, each kill a soldier makes is worth 30 experience points, or 60 if they are below Sergeant and it was an 'elite' alien, and enemy counts start from 4 enemies and go up from there. A soldier hogging all the kills in a big mission can easily get over 200 experience from that (On top of the mission completion experience), while everyone else is only getting 60 experience. Soldiers who just support their buddies and never finish off an enemy simply will fall behind.
As for that 'killed an elite enemy while at a low rank' bonus, this specifically means: Sectoid Commanders, Cryssalids, Berserkers, Muton Elites, Sectopods, and Ethereals. This bonus is, uh, kind of weird to be in the game? Chryssalids are honestly the main enemies a decent player expects to be killing with people below Sergeant: Sergeant is the third promotion, so only Rookies, Squaddies, and Corporals can get this bonus, and you really shouldn't be sending in such low-ranking soldiers in to fight Sectopods and the like.
Still, one of the more straightforward implications of all this bonus experience stuff is that the Newfoundland mission -Site Recon- is an experience bonanza for lower-ranking soldiers. As it's a very easy, very safe mission in which you can easily kill over 20 Chryssalids, there's a fair argument for trying to mostly send in lower-ranking soldiers so you can get extra value out of the pile of bonus experience. It's easy to have the entire squad gain a level.
Of course, there's the question: how much do these numbers actually mean in relation to ranks?
Annoyingly, the answer depends on if you're playing with Enemy Within enabled or running the base game. In the base game, it's:
Squaddie: 90
Corporal: 300
Sergeant: 510
Lieutenant: 745
Captain: 1100
Major: 1560
Colonel: 2150
Whereas in Enemy Within it's:
Squaddie: 90
Corporal: 480
Sergeant: 700
Lieutenant: 925
Captain: 1380
Major: 1840
Colonel: 2550
Additionally, note that Mecs halve the experience they gain from kills. Other experience sources are unaffected by this, so a Mec won't quiiite level at half speed, but just saying 'they level at half speed' is close enough to accurate: ideally you get people maxed, then convert them into Mecs.
Covert Operatives also get a really large bonus amount of experience. As far as I know, the exact details of this still haven't been dug up?
Regardless, if you want a summary: Enemy Within overall slowed leveling down, except reaching Squaddie is just as fast. The slowdown itself is slower at higher ranks: getting to Corporal takes 60% longer in Enemy Within. Getting to Sergeant takes slightly less than 40% longer. Getting to Lieutenant takes a bit under 25% longer. Getting to Major takes a bit under 20% longer. And lastly, getting to Colonel also takes a bit under 20% longer. (Relative to starting as a Rookie, to be clear: these numbers are slightly different if you instead focus on 'getting from X rank to the rank just above it', but I don't feel like calcing that whole thing; it's only kind of important to gift soldiers anyway)
Now, stats. Your poor Rookies all start with the following underwhelming statline:
Aim: 65
Mobility: 12
Will: 40
HP: 6/5/4/3 (Lowering as you go up in difficulty. This also doesn't count that all Armor gives at least 1 HP))

The Assault seems to lie somewhere between being your default line soldier and being a hyper-aggressive flanktastic murder machine that believes the best defense is a good offense. I feel the latter is the more compelling part of their design, aside how incredibly frustrating it can be to actually try to leverage it.

Squaddie
+1 HP
+5 Aim
+2 Will (On top of 0-6 random Will)
Run & Gun
Activated abilty: This soldier may fire or initiate Overwatch even after having used up all their actions in a turn, but otherwise cannot use activated actions of any kind once this ability is activated. 2-turn cooldown, cannot be activated when out of actions. Rapid Fire and Flush are allowed.
The defining skill for the Assault, granting them vastly superior ability to flank targets than other classes. Conceptually, I like Run & Gun as an ability. It provides a solid foundation for the Assault's role of "the soldier who runs up to targets and blows them away, preferably flanking them." by letting them pull off stuff your other soldiers just can't do.
I'm more mixed on the details of its implementation within the actual game.
First of all, a notable part of what makes the ability valuable is an AI design decision I really, really don't like: the AI in this game, as I explained back when covering Muscle Fiber Density, defaults to always having perfect awareness of all soldier positions and knowing exactly where they can walk to and fire from within a single turn. Run & Gun's biggest contribution to pulling off flanks is not actually the tremendous capacity to eat ground it offers, but is actually the fact that the AI doesn't 'see' it when planning its moves and so will aggressively move into a position that is impossible to counterflank with normal movement, but that Run & Gun can enable an easy flank on.
Second, this game's 'pod' system makes the intended usage of Assaults wonkily risky in a very nonsensical way. Pods only activate and become willing to freely move and fight if one of their members personally sees a player unit: a pod can be standing a bit back, able to clearly see the gunfire your troops are pouring into their friends, and it will refuse to activate and join the fight. But if your Assault comes running forward and gets spotted for even a split-second, suddenly instead of removing an enemy from the fight and making things easier on yourself, you've added 2-3 enemies to the fight and made it harder! Also, your Assault is frustratingly likely to be in a position to be immediately flanked and killed by such a newly-activated pod.
Like yeah, sure, moving aggressively makes sense to be a risky action, but this scenario is absurd and counterintuitive. It also gets exacerbayed by flaws with how maps are built and how the engine draws line of sight, where you can potentially make a Run & Gun move that really seems like it ought to have no risk of activating an unseen pod, and surprise, your Assault gets seen by an inactive pod anyway!
This whole thing gets particularly frustrating if you subject yourself to the game's higher difficulties, as one of the effects of going up difficulties is that more enemies get deployed into missions, including in the form of adding more pods. The game isn't very good at spacing out pods in general, and adding in even more pods makes that even more consistent of a problem, albeit in an erratic, unpredictable way.
Third, Run & Gun is... overly-restrictive on abilities once activated. If an ability isn't part of the 'whitelist' of shooting variants, it's locked off once Run & Gun is activated, even if it can be activated for free or feels like it ought to be logical to be able to use as part of a Run & Gun; this is particularly pertinent to runs with Training Roulette enabled, but in general it includes jank like 'cloaking with Ghost Armor is a completely free action that can be combined with Run & Gun, but only if you specifically cloak and then use Run & Gun'.
Fourth, a few enemies are inexplicably designed to punish getting in their face to kill them; several enemies explode on death for immediate damage, and most bizarre is Thin Men, who explode into a cloud of Poison on death. They do at least normallly get knocked away by the shot before they explode so a point-blank ahot isn't guaranteed to end up Poisoning the attacker, but if being knocked away would require entering terrain units can't walk in... then the cloud spawns precisely where they died and a point-blank attacker is Poisoned. I get that classic X-COM had some exploding enemies, so sure, those are back for faithfulness, but Thin Men are a brand-new enemy that's very common throughout a run all the way to very close to the beginning; what logic led to making them punish close-in combat mechanics? Why are we actively punching Assaults in the face with enemy design?
To be clear, there's not really any counterexample enemy designs that Run & Gun usage is stupendously favored against.No enemy specifically takes extra damage if shot from really close, or gets more Defense out of Cover so flanking them is extra-valuable, or anything like that. And other classes don't get a comparable issue of some enemy designs being extra-punishing to how they play, not in the base game; Enemy Within's introductuon of Seekers finally offers something, making Snipers sitting way back by themselves more ill-advised, but that's it, and it's very restricted; I don't get the impression the devs ever recognized they had enemies who were hostile to core Assault play in a way other classes don't deal with.

Corporal
+3 Aim
+2 Will (On top of 0-6 rabdom Will)
Tactical Sense
This soldier gains +5 Defense per enemy visible, to a max of +20.
vs
Aggression
This soldier gains +10% chance to crit per enemy visible, to a max of +30%.
I dislike both of these abilities. On an overall build level, one is the "defensive" pick and the other is the "aggressive" pick, but neither of them really changes how the Assault plays. An Assault with Tactical Sense doesn't play more defensively, nor does one with Aggression play more aggressively. In fact, neither of them really plays all that differently from the other, nor plays all that differently from an Assault that lacks either skill. Certainly, an Assault that has one of the skills has a higher average performance, but that's it. You're not going to change what kinds of decisions you make based on which skill you take.
It's also really difficult for me to argue in favor of Aggression in general. If you're going to take Bring 'Em On and Killer Instinct down the line, I guess Aggression is desirable for maximizing the odds of actually getting those crits you want... with three enemies visible, a Shotgun will always get a crit on a flank. But this line of reasoning is a bit questionable, as there are other sources of crit chance, and anything over 100% is a waste. A Council Medal of Honor set to the loner bonus can provide +10 crit chance, a SCOPE will eventually be upgraded by the Foundry to provide another 10% crit chance, all cloak effects give +30% crit chance, and Adrenal Neurosympathy adds 15% crit chance when its effect is active. An Assault in Ghost Armor or with Mimetic Skin can trivially get to 100% crit chance on flanks without any need for Aggression, and/or the Medal+SCOPE+Adrenal Neurosympathy combo can result in 100% crit chance on a flank if a kill was recently made. (Potentially by an entirely different soldier) All these options severely cut innto Aggression's value unless your goal is to get high crit chance in non-flanking cases. (Whether that's 'on targets that cannot be flanked' or 'you just want to stand back and volley shots from a distance')
And for that scenario, getting to 100% crit chance is... basically impossible. Hardened enemies in particular put you in the hole with -10% crit chance before you even start: a Shotgun+Aggression caps out at 40% crit chance on them, and then adding a SCOPE, the clook bonus, and the Council Medal of Honor loner bonus brings you at last to 90%, where you also need Adrenal Neurosympathy's recent kill bonus to push over the edge into 100%. For non-Hardened enemies you can drop Adrenal Neurosympathy, at least, but this is pretty demanding to actually get to max crit chance.
Then there's how the math of Defense is structured vs how crit chance works. Crit chance has dimishing returns as you climb: while it's cold comfort to be told 'that was unlikely' if a 90% crit chance does in fact fail to crit, adding in the extra 10% crit chance to hit 100% has literally no effect, well, 90% of the time. Defense boosts instead climb in value as you add more, up until you match enemy accuracy and the game refuses to let it drop below 1%. A soldier hiding in Full Cover when an Impossible Sectoid fires on them from maximum range has a 35% chance of being hit. If that soldier is a Tactical Sense Assault with the full +20 Defense, that drops precipitiously to a 15% chance to hit, erasing more than 50% of all the shots that would've hit; that's a big reduction in the number of shots that land on yuor Assault over the course of the campaign! Now, more elite enemies can have more accuracy, and enemies firing from up close get big accuracy boosts, but for example I'm ignoring the possibility that the Assault is getting Defense from other sources, such as by wearing a Skeleton Suit.
The only major qualifier to all this is that not every enemy action respects Defense in the first place, and more importantly that some enemy types largely or completely ignore it. Chryssalids, Zombie, Muton Berserkers, and Ethereals simply have no respect whatsoever for Defense, and Seekers will almost never fire their Plasma Pistol if they can initiate a strangle. Sectoid Commanders also heavily prefer their abilities, and Thin Men will usually respond to 'I can't draw a bead' by just spitting Poison. As such, there's an argument for building one Assault who does go for crits and deploying them into Terror missions, when going after the Overseer's ship, and anytime the Hyperwave Decoder is mostly stacked toward these enemies. It's not a good argument in my book, but it is an argument.
But yeah, I overall just default to Tactical Sense. Among other points, preserving soldier lives is several layers of useful; I'd rather stack the odds in favor of them living if I can.

Sergeant
+1 HP
+3 Aim
+2 Will (On top of 0-6 random Will)
Lightning Reflexes
The first Overwatch shot made on this soldier in a turn will miss.
vs
Close & Personal
This soldier gets increasing crit chance, max of 30% increase at point-blank, as they get closer to their target. (Enemy Within changes Close & Personal to: if this soldier fires a standard shot at an enemy from within 4 tiles of the target, the shot does not cost an action. Only triggers once per turn, and activating Run & Gun prevents it from triggering. Triggering it also disallows activating Run & Gun. A Rapid Fire is valid)
Lightning Reflexes turns your Assault into a linebreaker of sorts. Aliens playing defensively can be forced to waste their Overwatch on automatically missing the Assault, leaving your other troops free to move up in safety. Since Suppression is considered to be an Overwatch shot, it also makes the Assault essentially immune to Suppression, which is great when Aliens bother to Suppress you. This is an excellent skill, and I quite like the idea of it.
The original version of Close & Personal was terrible, and yet was a contributing factor to why Aggression is very questionable to take over Tactical Sense. If you really want crits, Close & Personal's maximum bonus, when stacked with the Shotgun's innate 20%, will turn a flank into an automatic crit. Since crit chance doesn't stack past 100% (It doesn't "rollover" into a chance for a double-crit or something), it probably makes more sense to take Tactical Sense and Close & Personal if you really care. However, I dislike the original version of Close & Personal on the design level, as it's both a very unreliable effect if you're not specifically using it to turn flanks into extremely reliable crits, and it does little to change the Assault's behavior. It theoretically encourages them to get closer, but getting closer is already incentivized by Aim mechanics if you're not talking a Sniper Rifle, flanks pretty much always involve getting very close (Thanks in part to how the AI prefers to move), and Close & Personal isn't a big enough boost to really be likely to influence exactly how close the player wants to bring an Assault to the target.
The Enemy Within version is actually one of my favorite skills in the entire game, as it provides clear, powerful benefits. It can be used to let an Assault get in even more damage, or to let an Assault get in, take a shot (Preferably from a flank), and then escape to a more defensible position before the enemy's turn, or take a shot at an enemy that's gotten on top of the Assault before fleeing to a safer position, etc. It also bolsters the Assault against melee aliens pretty consistently, so that's nice.
In the original game, I feel this is barely a choice at all: take Lightning Reflexes every time. In Enemy Within, Close & Personal is an incredible skill, but one that fills a very different role from Lightning Reflexes. If you want to straightforwardly make your Assault a more dangerous unit, Close & Personal is the choice to make. If you want your Assault to be a bit more of a supporting piece to the rest of your force, Lightning Reflexes is the better choice.

Lieutenant
+3 Aim
+2 Will (On top of 0-6 random Will)
Flush
Activated ability: fires a shot that forces the enemy to move from their current position (Even if the shot misses), which does 50% damage but has a +30 Aim bonus. No cooldown.
vs
Rapid Fire
Activated ability: fires two shots in rapid succession at a -15 Aim penalty. No cooldown.
Flush is frustratingly terrible.
It's tempting to suggest that Flush could go well with the Support and Heavy being able to build themselves as something of Overwatch specialists; have your Overwatch-specced Heavy go into Overwatch, then force your preferred target to move and trigger the Overwatch, so it certainly doesn't get wasted on the wrong target. This would even be a way to work around the infuriating jank where soldiers will all blow Overwatch on the same target at once, even if that's gross overkill and other enemies are about; that way, you could have a team with multiple Overwatch specialists where multiple of them can leverage that specialization in the same turn.
Unfortunately, Overwatch's design is incredibly frustrating and this use of Overwatch specialization is much worse than it seems like it ought to be. I'll be getting into it properly once we get to a class that actually specializes in Overwatch, but the summarized version is that an intuitive reading of the game's visuals suggests such Overwatch play ought to have advantages it mechanically doesn't have at all, and Overwatch is in fact normally just 'a regular shot, but worse', aside the potential for it to trigger during the enemy's turn on a target the soldier couldn't have shot at during their own turn. There's skill combinations that can make this firmly untrue, but... only if you have Training Roulette active and get lucky with distribution; a 'vanilla' run can't arrange this.
As such, this Flush utility has very limited relevance in normal play.
The alternate uses of Flush are more realistically existent, but still deeply flawed. Forcing an enemy to relocate really seems like it ought to be a great utility, but the actual behavior of Flushed enemies tends to not cooperate: you don't get any input on where the enemy goes, nor do they use simple and easily-predicted rules for deciding their location. Instead, they usually pick a location that's the type of position the AI would prefer to pick during its own turn: Flush will usually result in the target still being in Cover relative to your soldiers and indeed still unflankable by all your soldiers, and in fact can easily result in an enemy suddenly being in a positon to flank one of your soldiers where it had no flank options before you Flushed it.
Alternatively, maybe you're Flushing a target in hopes of it moving somewhere your other soldiers can actually shoot: this can potentially work... or it can result in the target deciding to run completely out of sight from your soldiers. Flush doesn't force the target to stay in vision or anything, and notably the game's line of sight/line of fire is at times asymmetrical for no clear reason, where an enemy can see and shoot one of your units even though none of your units can see it, so while Flushed units prefer to keep your units in view, that can just mean they move into one of these asymmetry points.
Similarly, it can be tempting to Flush an enemy in hopes that they'll move close enough to your Arc Thrower-equipped troops for you to get a capture, but this is... really unlikely to work out. You could alternatively focus on it as a way to do a weak amount of damage to soften up a target for capture, but a Pistol is overall better at that task, among other points having no risk of causing the target to move in a deeply problematic way. Once you've gotten the Aim-boosting Foundry Project for Pistols completed, Flush is only 20 points more accurate than a Pistol shot -and if your Assault is equipped with a Shotgun, Flush is at risk of being less accurate than the Pistol shot anyway by virtue of the target being far enough away for the Shotgun accuracy penalties to drag it heavily down. A Shotgun can in fact lose up to 40 Aim to distance, resulting in Flush being a total of -10 Aim while an upgraded Pistol would be at +10 Aim in that circumstance.
Similarly, it's tempting to suggest you could use Flush to force a target that's in Full Cover to at least move into Partial Cover so other soldiers are more likely to hit. Unfortunately, this game's maps are really prone to clustering Full Cover positions such that a Flushed unit will often have the option of just relocating to a different Full Cover position, and they are totally allowed to do that. There are points on maps here and there where a Full Cover position is isolated from all other Full Cover positions: for Small Scouts and Large Scouts, there's a decent number of cases of a Full Cover tree that's not near any other trees and not near any Full Cover rock formations. But this is overall a rarity.
Every once in a great while Flush's routine will 'break' and the Flushed unit will randomly shuffle to a location in the open, no Cover protection at all. Which, well, the AI sometimes does that in general; this isn't an intentional feature of Flush. And it really is rare: if you have Flush anyway and you've got no other options, I guess it's worth considering gambling, but selecting Flush in hopes of arranging this is... dubious.
This is also ignoring that Flush is pretty clearly unhelpful against all enemies who don't use Cover. It's also technically functional on Cover-using flying enemies (ie Floaters and Heavy Floaters), but they're very prone to responding to Flush by taking to the skies and so becoming unflankable. It's worth emphasizing that the 'endgame' collage of enemies has only Muton Elites as a Cover-using non-flying enemy of note: in the long haul, Flush gets dragged down by enemy compositions moving away from being units it's even theoretically decent against.
In practice, I'd argue the main value of Flush that works reasonably smoothly is to use it as a 'finisher': forcing a hit on a target that's low on HP and which you otherwise can't assure a hit on, so you can make sure it dies. And even there, well... this brings us to to Flush's biggest problem:
It competes with Rapid Fire.
Where Flush is a janky, flawed skill whose potential utilities are riddled with flaws, Rapid Fire is just... really good. It is of course straightforwardly amazing when making close shots at tough targets, where it can easily be at 100% accuracy even with the -15 Aim penalty, but it's actually really good even when making shots at shakier odds. Anytime a non-Rapid Fire shot would be at least 31% accuracy but less than 95% accuracy, Rapid Fire is mathematically improving your odds of landing at least one hit, and so even if you only need the one hit, it's still often better than just firing a plain shot.
Someone else has done the math, to support the prior, if you're curious.
Furthermore, Rapid Fire's ammo-hunger is less than you might assume, as it only performs the second shot if the target survives the first shot: thus, if you use Rapid Fire when you only need 1 shot, you don't even necessarily tear through your ammo faster. You may just have Rapid Fire be effectively a standard shot. And in the long haul, you'll eventually complete Ammo Conservation, at which point ammo becomes a bit of a joke; early in a run, mildly careless (or unlucky) use of Rapid Fire can genuinely put you in awkward positions, but later on you're probably being really sloppy if you get in trouble from Rapid Fire usage.
If Flush was competing with another 'eeeeh' skill, such as Aggression, I might argue in its favor, but competing with Rapid Fire is a big ask, and Flush simply doesn't deliver.
Enemy Within introducing Mecs with Collateral Damage further exacerbates the problems here. One of the only nice things one can say about Flush is that it's a way of trying to work around Cover that's freely renewable because it only spends ammo... which also describes Collateral damage, but it actually wrecks Cover so other units can reliably follow up and kill the now-exposed units.
Flush has more serious potential if you have Training Roulette enabled, at least, since it may actually get to compete with a mediocre skill. If the soldier in question is otherwise getting a great combination of strong skills going... then yeah, Flush might get a chance to kind of shine!
But by default? Ehhhh...

Captain
+1 HP
+3 Aim
+2 Will (On top of 0-6 random Will)
Close Combat Specialist
The soldier automatically performs a Reaction Shot against any enemy that moves within 4 tiles of the soldier.
vs
Bring 'Em On
This sooldier's critical hits do 1 extra damage for each visible enemy, to a maximum of 5 bonus damage.
Close Combat Specialist is an amazing skill, particularly in Enemy Within where Seekers are a thing. It won't activate on a Seeker trying to strangle the Close Combat Specialist, but it will activate if a Seeker goes for a nearby ally, reducing how many people you need to explicitly Overwatch. (eg allowing the Assault to simply reload, confident they'll fire regardless) The most obvious utility is letting the Assault get in free damage on melee enemies (Who will inevitably get in close to attack), but there are plenty of other situations where an enemy might move into close quarters with your Assault, and it'll be a free shot every time. (Unless you're out of ammo, anyway) The ability to automatically punish enemies turning corners unexpectedly, or to stand next to a closed door and be guaranteed damage on enemies if they come through it, and so on, is great studd.
Bring 'Em On is... ugh. There's several problems here.
First is of course that it depends on getting crits. While the Assault is arguably the best class at getting crits, this still often means you're gambling on crits occurring to get any value out of Bring 'Em On. This is especially problematic when fighting Hardened enemies, who are nearly impossible to get 100% crit against, and who become increasingly common as a run progresses, and include some of the nastiest enemies you most want to forcce damage on.
Second, its scaling depending on visible enemies is awkward, particularly the tuning. Needing 5 enemies visible to get the maximum bonus means you've pulled at least two pods, which you should generally be endeavoring to not have happen; if it was instead +2 to a maximum of +6, it would be straightforwardly strong as 'try to have your Assault make the first shot when engaging a new pod', but... nope.
Third, that scaling point has further awkwardness. Against a lot of enemies, it adds no value: a crit on a Floater splatters them unless maybe you were using a Pistol shot. Potentially critting for 11 damage with a Conventional Shotgun doesn't actually do anything. Alternatively, I can focus on how weapons technology drags down Bring 'Em On: for a Conventional Shotgun, +5 damage is equivalent to firing the Shotgun a second time and landing a non-crit that rolls max damage. For an Alloy Cannon, another non-crit would be 8-10 damage. As Bring 'Em On competes with Close Combat Specialist, this type of math is very pertinent: say you're up against a Classic Berserker, equiipped with an Alloy Cannon. I'll generously say your Bring 'Em On Assault has the maximum bonus and gets their needed crit, and further say they get the high roll result: alright, I made multiple favorable assumptions and your Assault does just barely get the 20 damage needed to instantly kill the Berserker.
The Close Combat Specialist also goes in and opens fire, and this provokes the Berserker into moving, resulting in a reaction shot that probably hits. 8-10 damage twice: this too can kill the Berserker if both hits roll high. If the first hit crits, this is in fact an assured kill, doing at worst 13 and then 8 damage for 21 total. Reminder that Bring 'Em On doesn't improve your odds of getting a crit: you hopefully did build for crits otherwise if you took Bring 'Em On, whereas you might not have if you took Close Combat Specialist, but you really do need to lean into crit chance boosts to make Bring 'Em On seem appealing.
Now, if you take Rapid Fire and Killer Instinct, Bring 'Em On can potentially result in wacky-high damage output, where your Assault charges in, shoots twice, and crits for 30 damage each time. That's enough to kill even an Enemy Within Sectopod in one go, assuming you got both crits and rolled high on both, which is... a pretty big pair of qualifiers.

Major
+3 Aim
+2 Will (On top of 0-6 random Will)
Extra Conditioning
The soldier gains +1, +2, or +4 additional HP, depending on their currently-equipped Armor.
Makes the Assault somewhat tougher. Enemy Within provides a notable change by making Carapace Armor qualify for the +4 bonus, instead of the +2 bonus. Outside of that exception, it roughly correlates to tech level -you get +1 if you have nothing but the basic Body Armor equipped, +2 for most more advanced armors, and +4 for Archangel and Titan Armors in particular.
The effect is pretty small, and while it slightly biases the Assault away from Ghost Armor as their endgame armor and, in Enemy Within, gives them reason to consider sticking to Carapace instead of moving to Skeleton Suits the instant you have them, the effect it has on armor choice is also just not that big. It might keep your Assault alive one shot longer, but it's also entirely possible for the bonus HP to be irrelevant in the face of Alien firepower, particularly in the late game. An Assault in Titan Armor on Impossible, for instance, has 21 HP instead of 17, which will only occasionally make a difference against an Impossible Mechtoid's base damage of 11. (It could roll low twice in a row and thus only do 20 damage, but it usually won't)
I do appreciate that Enemy Within made Carapace get the +4 bonus, though. It means that early-mid-game high-experience Assaults are actually noticeably more durable than your other classes, aside from the somewhat complicated comparison to Heavies. (Also, Mecs win the day period, but they're much more expensive than an ordinary soldier and are sufficiently different that direct comparisons don't work so well) I personally feel the Heavy's equivalent skill is far, far superior, no questions asked, which is a bit disappointing, but it's not ruinous to game balance. It's annoying, particularly if you're running Training Roulette, but it could easily be worse.

Colonel
+1 HP
+4 Aim
+2 Will (On top of 0-6 random Will)
Total stat gains
+4 HP (For a total of 10/9/8/7 max HP, based on difficulty)
+24 Aim (For a total of 89 base Aim)
+14 Will at minimum, up to a maximum of +56 Will (For a total of 74 Will at minimum, 134 Will at maximum)
Resilience
Attacks on this soldier will never be critical hits.
vs
Killer Instinct
Activating Run & Gun adds 50% damage to critical hits for the rest of the turn.
I personally prefer Resilience, as Killer Instinct is only fully reliable a benefit if you stack on critical hit boosts and grab a flank (You'll need +30%, even with a flank and a Shotgun, to reach 100% crit chance), and one of the problems with trying to get a flank, even with Run & Gun, is that you don't necessarily have a flanking position on your target that keeps you in Cover, and indeed running into the middle of a group of enemies means that often even when you do grab a position that's currently safe, multiple enemies will be one move away from flanking you anyway. As such, Resilience frees the Assault up to actually get flanks consistently without worrying about being torn apart by crits, where taking Killer Instinct can put you in a position where actually taking advantage of it is virtually guaranteeing the Assault's death. If you play more conservatively with the Assault as a result, you're even less likely to get crits in the first place, and therefore Killer Instinct becomes somewhat self-defeating of a choice, given that Resilience is what puts you in a position to actually take advantage of Killer Instinct's benefits -but you can't have them at the same time! I really think Resilience should've been the Assault's Major skill, and then Killer Instinct placed as competing with something like Lightning Reflexes or, heck, Extra Conditioning.
I like the idea of the choice, that you're choosing between emphasizing the Run & Gun flank effect by virtue of not having to worry overly much about keeping the Assault safe vs emphasizing the Run & Gun flank effect by making it even more murderous, but I don't think it really works out, especially since the further you get in the game the less common flankable enemies become. Muton Elites are basically the only end-game enemy that you can consistently expect to flank -everything else is Hardened and/or Flight-capable. That makes Killer Instinct even more unreliable and limited the further you go in the game. The only point in Killer Instinct's favor is that it finally provides a big payoff to investing in the Assault's +crit skills: with the original version of Close & Personal plus Aggression plus a S.C.O.P.E. with the Foundry Project upgrade plus the base crit chance from Shotgun-class weapons, an Assault can get to 90% crit chance without a flank. With Bring 'Em On and Killer Instinct and Rapid Fire, a crit-focused Assault wielding an Alloy Cannon can do 15+5x50%x2=60 damage. That's enough to one-round a Sectopod in Enemy Within!
On the other hand, that's the best-case scenario, and demands the impossible scenario of using the Enemy Unknown version of Close & Personal against the Enemy Within version of the Sectopod. In actuality, either you're playing Enemy Unknown and that's hilarious overkill against basically everything in the game, or you're playing Enemy Within and you just can't get reliable enough crits to make this an effect worth counting on. Enemy Within's additional items and Gene Mods and Medals don't make up the difference of the loss of Close & Personal's crit chance bonus: even with Depth Perception's bonus, the pheromone boost of a recent kill, and +10% crit chance from the loner version of Council Medal of Honor, these only add up to +25%, instead of Close & Personal's up to 30%, and they're also all a lot less reliable to pull off and, in the case of the Medal, require you didn't pick the other option. A more realistic upper limit on the Assault's crit chance in Enemy Within is 60% -and then you drop it to 50% against Hardened targets like the Sectopod. At that point you have about a 25% chance of actually having your Assault vaporize the Sectopod, and about a 25% chance of getting zero benefit from all your crit-based skills. That's godawful, not remotely worth planning around. Even if I assume Ghost Armor, Mimetic Skin, or a Ghost Grenade is used to bolster that by 30%, it's still shakier than I'd like, especially since the value I'm citing for one-rounding a Sectopod with a single Assault additionally demands both shots roll their higher critical damage number, making the one-round kill even less likely than I'm presenting it as being.
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So I mentioned when covering Tactical Sense vs Aggression that Tactical Sense is the "defensive" trait while Aggression is the "aggressive" trait. That dichotomy basically defines the Assault's two "lanes" -the left set of skills is all "defensive" skills, and the right set of skills is all "aggressive" skills. Unfortunately, this is a case of the remaquel falling into a trap many games fall into -that of thinking that enhancing defensive ratings is automatically a boost to defensive play, while enhancing offensive ratings is automatically a boost to aggressive play. In actuality, it's more nuanced than that, and at the extreme ends it tends to flip around. If you're literally invincible, you can be as reckless as you like because you can't be hurt. It's when you're vulnerable that you have to play carefully.
Compounding this is that there isn't really such a thing as "defensive" play in the remaquel, not for the player anyway. There's cautious play, where you move slowly and as often as possible end your turn with all your soldiers in Overwatch in an attempt to avoid activating more than one Alien pod at a time and ensure that you'll get to reaction fire them as they activate and then get a full turn on top of that, but the XCOM Base Assault mission in Enemy Within and one form of EXALT mission are the only times where hunkering down and waiting for the enemy to come to you is particularly viable as a way to win a mission. (Keeping in mind that the XCOM Base Assault happens once per game, while EXALT missions are both far too easy and stop happening once you've hit EXALT's HQ) This is compounded by the fact that some Alien pods will simply never move from their position until activated -mostly this means the "commander" pods on UFOs (Outsiders, Sectoid Commanders, or an Ethereal with Muton Elite bodyguards, no exceptions) but even non-commander pods will sometimes just sit around and do nothing indefinitely.
However, even if I decide to call the cautious form of play "defensive", none of the Assault's "defensive" skills actually enhances their effectiveness at cautious play. At most the "defensive" skills are less strongly incentivizing reckless play than the "aggressive" skills, which isn't the same thing as encouraging cautious play.
Worse, the Assault's skill pairs just aren't well-balanced. Lightning Reflexes vs Close & Personal is the closest to a balanced choice, and even then only in Enemy Within. The rest of the pairs are weighted so strongly toward one skill over the other there's little reason to consider more than the one Assault build.
I consider the Assault to be one of the more poorly constructed classes, which is a bit odd, as they have the best-designed, most-class defining Squaddie-level skill of any class except maybe Collateral Damage on Mecs. A strong start, but a bland everything else. Their class-unique weapon of Shotguns does little to help define them since it's a slightly more extreme version of the game's default assumption (Specifically: getting closer is good for your Aim), and they have no other intrinsic properties to really set them apart from the other classes. Some of their more advanced skills help clearly place their role -Resilience and Killer Instinct both drive home the Run & Gun flanking murdermachine role, albeit in slightly different ways- but they don't particularly manage to carve out multiple different kinds of niches for Assaults, excepting the Enemy Within choice of Lightning Reflexes vs Close & Personal.
Next time, we cover the Sniper.