Well, I don't play Terran but I'll chip in as regardless of faction, the differences between a carrier and a non-carrier will still be clear enough even taking into account the fact that one of them is an Asgard.
So, principally, one is a carrier and one isn't. Seems like it's just a word, but it matters. Whilst they both can house 40 S class ships, the Tokyo has 18 actual docks for them, vs the Asgards 3. Meaning that when the shit needs shoveling, the Tokyo can launch the whole lot in 2.2 waves, vs the 13.33 waves needed for those same ships to launch from the Asgard.
Lets say that it takes 30 seconds to launch a wave (guessing, but it's probably fairly close), the Tokyo has everything out and ready in just over a minute, probably less considering that the first wave will be on the pads and ready to go anyway, vs the almost four minutes needed for the Asgard to ready its fighter complement.
If you're running with flight wings of torpedo bombers, that's enough of a time difference to destroy 3-5 Xenon K's should they have tried to ambush your capital ship. Of course, the Asgard has its Bullshit-CannonTM to help it in this respect, but even that isn't taking out 5 K's in under four minutes due to the recharge time. Of course, sending in either class of vessle without an escort fleet would represent an almost criminal negligence of acceptable fleet doctrine anyway (normally, there are edge cases though). In real terms, there is an issue of only having a wing of 40 torpedo bombers, meaning that they may run out of ammunition before all 5 K's are dead, I can't say that I've ever been ambushed by 5 at once so I'm not sure exactly how many one wing of 40 can take out. I know that 30 Chimera's setup that way can take out at least 3 without running dry though, so I'm confident enough about the numbers. I'm not sure what the Terran equivalent ship to the Chimera would be, pick a heavy fighter and you're probably good to go.
There's also the question of wether or not your capital ship can even survive for those four minutes under that kind of fire... Much would depend on how good the pilot is at tactical maneuvering, etc. I've not ever actually used an Asgard, but I'm sure they have some survivability here, they're almost legendarily tanky so far as I know.
Yes, carriers are massively OP if you have all of the fleet logistics and support industry setup correctly, even moreso than the Asgard.
Of course, getting amubshed by fleets of K's isn't exactly the standard use for a carrier/battleship, so what about the difference between the two in more normal situations?
The main thing here is that because the Tokyo is an actual carrier, it has access to all of the carrier AI that comes with one, such as Position Defence, allowing you to deny specific areas from enemy access, setup rapid response forces from a central location that will strengthen other areas on an as-needed basis, etc. This is all remarkably useful and cannot be done from a non-carrier platform regardless of how much ship storage space there is. It all adds into the actual worth of running a carrier group, which is force projection over an extended area, whereas an Asgard is about being an unstoppable removal tool against specific targets.
Also, as a carrier the Tokyo can automatically repair docked ships, ships which will automatically re-dock when combat is finished when using normal attack commands, or re-dock on an as-needed basis when using Position Defence points, etc. An ability which will greatly increase the resilience of your fleet when on a protracted deployment. None of that is going to happen with an Asgard as it isn't a carrier and has no capacity to do any such things.
Of course, unless you have specific reasons such as doing a Terran Only run, you should just use a Raptor instead of a Tokyo, but you're literally doing a Terran Only run, so...
The carrier specific things going on in the Tokyo (and all XL carriers) make it effectively a mobile airbase, with all of the maintenance facilities and command and control capacity which comes with that. When considered for the same purpose, the Asgard is merely a large box that can carry a bunch of ships. That's your main difference there.