Final Fantasy XIII-2 may honestly be the only approvable thing Square Enix has managed to pull off since their inception as a merged company. Where FFXIII had plenty of flaws and unwelcome changes, XIII-2 manages to implement a nearly identical system while correcting every last problem with the original game.
It could just be me, and the fact that the story in XIII-2 bears many thematic similarities to Chrono Trigger, but the game is enjoyable to the point where you almost forget it's part of the Fabula Crystallis Novis disaster.
Admittedly, the story makes absolutely no sense, not to mention that right off the bat they dropped the retcon bomb and they painstakingly remind you of that by showing you, over and over and over, modified or extended versions of the end CG movie from FFXIII. At least it makes no sense at first, though I'm going to stick with it and say that it still is kind of out there even with the expalanations.
Gameplay:
Most importantly, Squenix resolved the single biggest design failure in XIII: If your party leader dies now, it is not game over. You simply switch to your other party member (you only have two + a monster) If they both die then yea, you lose, but that will never happen. They also fixed one of the other biggest problems: You can now go wherever you want whenever you want (provided you have unlocked the timelines).
They also have made the crystarium far less absurd. While you still have 8 million nodes to unlock, it costs signifcantly less to do in this game (think like 100th the cost). The only problem with it is that you don't control which nodes, but you are going to unlock them anyway so it doesn't really matter.
FFXIII-2 has extra things to do, unlike it's predecessor. To an extent, anyway. The classic chocobo races have returned, and there's a casino as well. On top of that, you have side-quest "fragments" to collect, and artefacts, which are mandatory if you want to get 100% completion and the good ending.
Combat system:
The combat system is exactly the same, excluding the improvements mentioned already. Now, however, you have two humans, and a monster. You can switch your monsters out just like characters, and so you use a different monster for each role - though they stupidly share a health bar.
I do believe there is QTE included now, too, but it's not really important.
Story
As I said the story really makes little sense. They retconned the goddess Etro into the story and so it has changed the game environment a lot, not to mention introduced all these new elements that just are weird. As the game goes on, though, it unravels, and makes you forget how nonsensical your quest to not do what you're going to end up doing anyway was in FFXIII.
Also, if you ever wanted to hear the death metal version of the chocobo song, this is your chance.