Final Fantasy XIII had its fans. I was among them, and I've taken a fair amount of stick for that. But even I, in my Pulse-induced state of delirium, would admit that it had issues. Taken as a riposte to those complaints, XIII-2 is something of a petulant over-reaction. Taken as a sequel, it's a pleasant echo that's lost a little nuance. Taken as a game in its own right - something that's almost impossible for a Final Fantasy game - XIII-2 is pretty damn good.
In terms of story, it's great - you'll find out by what deus ex machina the cast of the first game escaped their certain fate. You'll be satisfied by the dire and slightly incomprehensible consequences of that intervention. You'll nod your head and smile through the babble, because Final Fantasy XIII-2, like so many of these things, boils down to a complicated and unbalanced love story between people with far too much power.
Complaint one from the first game: too many awful characters. Snow was one of the most likeable, and even he was oblivious to the point of facepalm. Sazh was alright. If I was a Chocobo chick, I'd have nested in his hair, too. But his adorable son could eff right off.
The cast problem has been totally fixed, thanks to the fact there's just two main characters. Serah runs ridiculously, using the same animation as the skin-crawling Vanille. But she's no gasping woman-parody. She's reasonable, talkative, and likeable.
Noel, our companion from the future, is neither moody nor hyperactive. He's a guy with issues, but he's not a miserable bummed-out douche about it. Of course, there's an irritant. Your fully-voiced Moogal companion (Kupo) does his best to ruin the atmosphere (Kupo) by saying Kupo all the time (Kupokupo), but even that doesn't kill this new, amiable atmosphere. You even learn pretty quickly how to throw him off a cliff. He comes back, but it's therapeutic.
The most common criticism was that the first 15 hours of FFXIII were a long, long corridor. The surprising reaction to that is the Historia Crux. The Historia Crux is your new corridor, a menu screen with discrete locations across time and space spread among its branches. There's no physical connection between the maps. You unlock new locations by delivering artifacts to gates, opening new locations on the Crux.
The corridor of the first game was a long, long tutorial - pairing the six characters off in every possible combination to make you learn the interactions of each of the six combat roles. A certain familiarity is assumed, here, so that corridor wasn't necessary. However, this new solution gets rid of the world. There's no arrival at Gran Pulse. No moment when you're given an airship. It may be functionally the same as having a world, but there's no sense of expedition. It's a decision that is going to ruffle a lot of phoenix feathers.
The shops of XIII were a single, impersonal terminal for everything, which meant cities disappeared. Cities aren't back in XIII-2, but we've got ourselves a shopkeeper. Welcome Chocolina - your hyperactive and surprisingly inoffensive Chicken Lady, who pops up everywhere in time and makes no bones about wanting your cash. She's your upgrade shop, too - although accessories and upgrades have been pruned back to a stump. FFXIII had a genuinely unlikeable, opaque and artificially complicated upgrade process. By contrast, XIII-2 is openly shallow. You'll pick up items from monsters that you can use to upgrade existing stuff, but only to make them more powerful and costly, or weaker and less costly. (How much you can wear is limited by a "cost". You won't be able to wear really powerful items until you boost your accessory stat.)