Well, I could sort of just tell you that Kingdom Hearts 2 is basically an improvement and refinement of KH1 in just about 99% of what it does, but that would be short and boring, wouldn't it? Not to mention that it wouldn't really tell you anything. So instead I'll be longwindedly boring, and talk about the improvements in detail. I think such a comparison makes sense, since after all, it is a real sequel to Kingdom Hearts 1, and you should be familiar with that game plotwise if you plan to play this. It can get really confusing otherwise.
But before we get to the improvements, I'll have to mention the main frame from the first game, which is pretty much the same as it was. Your three heroes Sora, Donald Duck, and Goofy will travel in a fairly linear manner between worlds mostly known from Disney's animated movies (and loosely re-acting the plots from the movies as well, in most cases), meeting a hell of a lot of both Disney and Square characters along the way, and of course getting a lot of challenges, mostly in the form of combat. Apart from that, all I will reveal about the overall plot is that you really need to have played the first game to get the most out of it; and even so, many things in the main plot can be rather confusing for a good while. But it is also at times more epic in scope (it's more than just Sora's own struggle this time around), although I wish this could have been a bit more often, because those are definitely the best parts.
Now, the core combat gameplay - which is what you will mainly be doing when you're in control - is as following. You walk around and fight monsters in real-time by controlling the hero Sora, who wields a magic sword that looks like a giant key (hence the name Keyblade). At the beginning, all you can do is pretty much a three-strike combo, by continously pressing X. You also have two companions (usually Donald and Goofy) whom you cannot control directly, and whose main job is to draw some enemy fire and to heal you when your HP is low (because if you lose all your health, it's Game Over. If they lose their health, they're just incapable of doing anything for a while). Now, in KH1, it didn't proceed much beyond that, but here, things are different.
First of all, the Triangle button has been given the task of "reaction button". Taking a clue from Zelda: Wind Waker, when the enemy is about to do a specific move, you press Triangle and thus counterattacks said enemy. And this is the really cool part: Each and every one
of them has a different reaction counter (with its own name), which means that you're looking forward to meet every new enemy (which there are in fact a good deal of) just to see what cool new trick you'll be using against them. And not all of them are strictly counters either, some of them are stuff you will be able to as an ending to your combo, which of course hurts a lot extra. Some of them can even hurt more than one enemy.
This alone makes the combat system so much better than KH1, especially against certain of the bosses, whom sometimes a well-triggered reaction can lead to some seriously awesome-looking smackdown. For that matter, the bosses are in general several notches up in variety from how you dealt with them in the first game. They are all still possible to beat, but if you think that even Proud (Expert) mode will be the cakewalk it was in the first game, you might be surprised...
And that's not all. Because not only do you have the reaction button, and not only do you eventually learn a hell of a lot of ability moves that improves the basic fighting more than it did in KH1, but you now also have Limits, improved Summons (which even have their own limits), and Drives. Limits is basically that you act with one of your companions to unleash some serious damage (there's at least one limit for each playable character). They usually take the form of a series of small, controllable attacks, and ending in one big attack with a bit longer animation (but never so long as to get boring) Summons can actually be useful this time around, though it's still possible to beat the game without them. And as for Drives... Well, the gist is that you lose the help of either Donald, Goofy, or both, but in return, you turn into a real killer. How about wielding two Keyblades at once, letting a rain of swordblows down on your enemy? Or maybe turn your Keyblade into a gun and effortlessly glide around on the ground while shooting your enemies to smithereens from a relatively safer distance?
And if you want to, you can even make it so that the reaction button will act a shortcut to either of those three options when you're closing up on the enemy. Oh, and speaking of shortcuts, when you press and hold down L1, not only can you customise the face buttons to do some magic (like you could in KH1), but also to spend items, so that you won't have to do that cumbersome item selection thingie which can be hard to pull off in the midst of battle frenzy. And
you can set it so that as soon as one battle is over, you'll automatically pull the item you just used in said battle and in the "battle slot", meaning you'll never have to go to menu each time you had to spend a Potion because you got hit a bit too often from the big bad monster. Small but important detail there.
There are dozens of other small details omitted from my descriptions, but the point is this: The combat system works much better this time. Yes, it can still get a bit repetitive when you're doing all that leveling up (did I mention that summons and Drives have separate levels? At least they actually level up a bit differently, though not much), but at least there is some actual variety around. Especially the boss battles (I've said that before, haven't I?)
Another move which they pretty much took directly from Zelda: Wind Waker is the improved camera. You can make it stay directly behind you and change view as your character does, or it can stay at a put angle and altitude, which can both easily be adjusted with the right-hand stick. The only (small) downside to this is that since the right-hand stick is now used for the camera, it can be more difficult at times to use any sort of battle command that doesn't have a shortcut, especially in the kind of battle where you constantly have to move around. But that seldom happens, so most of the time, I consider this a good trade-off.
Outside of battle, the triangle button keeps doing its job as the reaction button, meaning that you use it to talk to people, open treasure chests and push levers and such. As soon as you get used to it, you'll realise that this works better than having the main command on the menu in the lower-left corner change into "talk", "Open", etc, because you never know when you might need to attack something.
The second most noticable improvement is how you move between worlds. The gummi ship is back, but with several changes. First of all, as long as you have access to a world, you can move the gummiship freely around, and it's much easier and faster to choose your landing point. And then, there are the parts when you have to open up new paths...
Instead of the miserable shooter-wannabe that looked like the original Starfox for the SNES, we are now treated to an actually nice rail-shooter with compelling enviroments, that can hold its own. Not only is it much more fun to simply open up new paths this way around, but you can also afterwards engage in up to three different missions around the course you opened; which will allow you to earn plenty of new stuff (especially new weapons) to add to your gummi ship. These missions are 100% optional, but whenever I needed a bit of rest from the main game progress, this was what I ended up doing. If you don't find at least "Ancient Highway" and "Ruined Highway" somewhat exciting, you probably don't like rail shooters at all.
The gummiship editor is not improved in any significant manner, still being initially confusing, and with a hopeless "guide" that just gives you a headache to read through. But once you finally get the hang of it, it's not that hard, but it seems to be a bit too complicated considering it's not important to the main game as such. However, as long as you only use the gummi ship for opening pathways, this won't matter much, because it's easy to just acquire ready-built ships that will at least be adequate for the opening of paths. Of course, if you really want to succeed in the missions, you still have to learn to build a ship from scratch, because you will need serious firepower for some of them...
Graphically, it looks very much like the first game, which works just fine. There are improvements here as well, but they are mure subtle, at least at first. Some worlds are rather dull, while others are quite the sight. As for the music, it's high-quality all the way through, always fitting for the worlds; and the song in the opening and ending is strangely fascinating, and about as good as I think j-pop can ever be. And with Disney pulling out from their large pool of voice acting talents, as well as hiring up some top-notch actors like Cristopher Lee, there's a bigger chance you'll cringe from some of the silly material the script-writers gave them, rather than their performances.
So, if you enjoyed Kingdom Hearts 1, there is a 99% chance that you'll enjoy this even more. They made most of the boring stuff good, and most of the good stuff better. There are some surprises hanging around, and I haven't even mentioned all the small and silly minigames scattered around (not that many of them are particularly good, but there are many of them). The pacing can still at times be a bit slow, the different world plots seems to be a bit detrimental to the main one (which is the best one), there seems to be more moral lessons happening than in the first game (bo-ring), things can be a bit too confusing, and the worlds can still feel a bit empty and devoid of people... But it's nevertheless a very solid experience overall, and my B+ is close to approaching A-.