Aspects are true. Boosts aren't, really.
Just to restate the full text of the stunts you're talking about, you trade a Boost for an Aspect with a free invoke on it:
Heavy Hitter. When you succeed with style on a Fight attack and choose to reduce the result by one to gain a boost, you gain a full situation aspect with a free invocation instead.
from Skills & Stunts: Fight
and a boost is pretty much a free invoke already, it's just a "free invoke" on nothing in particular. You can't compel based off of boost text and there's some suggestion to just write down "Starhound's Shoot boost" if you can't come up with something.
Whereas if you clock someone with Heavy Hitter and they get Staggered, well, they're staggered. Even after you spend the free invoke, the Aspect doesn't go away unless it would make sense for it to in the plot - which in this case means that the person you staggered, or someone helping them, will need to Overcome Staggered before they can, say, run full-tilt away from you or whatever.
Defining the Aspect
Let me preface this by saying that it's extremely difficult to talk about Fate in the featureless white room of theorycraft. You have a lot of judgment calls to make as a Fate GM: is this too easy, too hard, or "let's roll"? Is the opposition to it active or passive? If it's passive, how high is the passive opposition? The state of play is going to inform a lot of these decisions, but as a general guideline, an Aspect created as a side-effect of a stunt should do something, not nothing, but also not everything.
When you roll Create an Advantage against a low passive opposition, often what you create is an aspect that "does nothing" - grants no special additional permissions but has a couple free invokes and can be hit with Fate Points in a way you might not currently have access to. Dark Stobolous can roll Lore against a low passive opposition to meditate on the black hole that powers him and gain the Aspect Child of the Schwarz, probably with a couple free invokes because his rating's high. Mostly what this does is kick up some gravity SFX in the environment, which might slightly extend the reach of his active opposition, but the main benefit in addition to the free invoke is that he can double-kick Child of the Schwarz and Powered by a Black Hole with Fate Points to put up the big numbers with gravity.
When you Create an Advantage against active opposition, you have license to make an Aspect that "does everything"; for example, if Athens gets the drop on one of Dark Stobolous's goons she could roll Fight or Physique to Create an Advantage like Blade to the Neck or Submission Hold that would credibly prevent the goon from trying to do anything other than break out of it with an Overcome. They wouldn't be a totally helpless target; anyone from outside would still have to be careful not to hit Athens.
In a Conflict, "something" is going to mean, pretty much, "block one course of action". It's very rare than a Conflict in Fate is going to be only two sides whacking numbers at each other until somebody's numbers run out, but it's also very rare that it involves large portions of freeform action. So the easiest way to model an imposed Aspect is that it will restrict the target in some way - Staggered will let them attack and defend, but not make significant movements or block someone else's maneuvers. Disarmed means they lose their weapon rating and associated stunts and aspects to Shoot or Fight (or, if you're not using weapon ratings, it's likely to be a "cinematic disarm" that leaves them unable to Attack an armed opponent).
Dealing with the Aspect
Clearing the Aspect is almost definitely going to "waste someone's turn" in the exchange, and can happen as a result of an Overcome or Create an Advantage action against passive or active opposition, depending on how actively you want to involve things. There isn't really a hard and fast rule here, other than mooks are usually going to be limited to Overcome - real threats have it in them somewhere to Create a complementary Advantage.
As far as rolling against, it depends what you did in the first place - Staggered isn't something you can actively enforce, so breaking out of it is passive opposition, and the source of the difficulty is you having clocked a guy, so the passive opposition equals your Fight score. If you hit someone hard enough to Disarm them, then you can actively enforce the Disarm by rolling... probably Athletics or Physique, to keep up with the scramble for the weapon or block their way. This doesn't expose you to any side-effects of a poor defense roll other than losing the aspect.
Of course, if you stagger or disarm Dark Stobolous he's just going to roll Lore to create the Advantages Child of the Schwarz or Extruded Nictic Claws - against passive opposition in both cases, based on your Fight in the first case or... probably minimum of Provoke or +2 for the second, since what gets in the way is both the difficulty of the technique and how much you rattled him by knocking away the Fission Tulwar? (Also he'll still kinda be disarmed, it'll just be "Disarmed (Fission Tulwar)" and he'll lose access to its special powers but still be able to mix it up just fine.)
Better Than New is a little tougher to conceptualize in this way, because a Souped-Up Engine or Reinforced Hull isn't actively blocking people the same way being Staggered or Disarmed would, and probably isn't being created in a Conflict in the first place. So you have a little more leeway to give it that "something". (Also since Overcome is a prime Challenge action, the next person on is using the thing you fixed and can vector a Fate Point off your repairs when they couldn't off of you.) It's easier for Twilliam to dive for cover inside the ship if it's got a Reinforced Hull, for example. Or, when Imperials are chasing them through the fog-swamps, that Souped-Up Engine lets Starhound try and create a Lost in the Mist advantage at a lower passive opposition or perhaps even at all.
Best Foot Forward is limited to twice a session because instead of being linked to a specific action result (succeed with style on an: overcome with Crafts, defend with Athletics, attack with Fight) it's any time you get a boost with any action, even the times when you get one as a result of a tie. So Twilliam can, for example, put "I Like Your Moxie, Kid!" on a crime lord and that lets Starhound leverage Resources to get supplies at the black market, or put Thick as Thieves on a spaceport guard to remove him as active opposition when Athens is making some extremely not-street-legal modifications to the landspeeders. (There is still a low passive opposition, representing task difficulty and the general public's tendency to be a looky-loo.)
Backup Weapon is a little misleading in this regard because -- well, first, because it's assuming that PCs are operating in a setting where the specific aspect Disarmed means something, instead of nothing, to all of them. Second, it's not actually about a special conversion, but forcing a specific mechanical result - someone has to Create an Advantage to make you Disarmed in the first place, right? Well, instead of spending Fate Points on the defense roll, you can just make it, and then spend a Fate Point to turn it into a tie, no matter how badly they beat you. Even then, this one's a little in the old model of "stunts indicate to the GM how you want your character to be played to" - I'd suggest you could also apply it to any disabling Aspect a hidden weapon could help with. Tied Up? Knife up the sleeve. Held at Gunpoint? Back holster. Glued to the Wall? Not by my other hand I'm not! That kind of thing.