Let's get this out of the way first: summons absolutely sucked in Final Fantasy XIII. They were severely underpowered and difficult to use, to the point where it really wasn't worth using them at all. Rather than fixing that broken system though, summons are dispensed with in XIII-2 in favor of a monster collection system. Since Serah and Noel are the only two human party members in the game, the third slot in battle is filled by your choice of beasts.
Above: Some monsters are cuter than others
As you defeat enemies, they often drop a crystal that adds the monster to your bestiary, allowing you to add it to your battle team. At any given time, you can choose up to three monsters to use as the third slot in your Paradigm deck. Since each enemy has a set role (Commando, Ravager, etc), choosing three roles that complement the rest of the team is essential. It's not just about strength, but balance too.
They're not all winners, but separating out the chaff comes with the territory when you're talking about monster collecting. The weaker monsters are still useful though, because instead of just setting them free, you can "infuse" them onto the monsters you actually use. This adds passive abilities to the monster that absorbed it, like elemental defense boosts and increased gil drops, so no monster ever goes to waste. This definitely means you'll be spending a lot of time sitting in menus if you want to really maximize your optimal set of monster allies, but if that's really not your thing, you can put it off and make due with average monsters too.
Above: See the moogle in action in this PS3/360 comparison video
So, everyone is happy that moogles are back, right? Well, the adorable Mog does hide one major change in FFXIII-2: the return of random encounters. The annoyance of these random encounters is mitigated by the Mog clock, which allows you to run from battle if you prefer, but we still miss the visible enemies of XIII. It's a concession we're willing to make in the face of technical limitations though, since the tradeoff is more open environments to explore.
On the plus side, Mog's presence adds a much-needed light-heartedness to XIII-2. Where XIII was humorless and took itself way too seriously, XIII-2 isn't afraid to poke fun at itself here and there. Of course, it still has its moments of serious drama, but now they're broken up with bits of silly dialogue here and there that really help lighten the mood.
Final Fantasy XIII? Yes, in the sense that most longtime Final Fantasy fans will enjoy it more. It's definitely a tradeoff of pros and cons, but the single bullet-point of non-linear exploration will tip the scales in FFXIII-2's favor for many people.
White Knight Chronicles 2? Yes. This shouldn't come as a surprise, since we didn't care for White Knight Chronicles 2 at all, but if you were really wondering then there's your answer. Yes. It's better. Much, much better.
If FFXIII was an exercise in form over function, its successor is exercise in working with what you've got – and doing a brilliant job of it (for the most part). It does some things better than its predecessor, some things worse, but what's most impressive about Final Fantasy XIII-2 is how much feels new and different, despite sharing most of its guts with XIII.