I say yes. Here's why:

The Battles
Better by default, since it's an enhanced version of the same thing. I personally don't care for this system because I prefer slower battles with more control over character action, but I can't argue that this is improved over the original.

The Characters
FFXIII's cast was stronger and deeper, so it's hard for a two person cast to top that, especially when one of them is Serah. However, I think Noel was a great character and up there with any of the FFXIII six. Caius with his overly-dramatic voice and cheesy dialog was still a far better villain than crappy Barthandelus. As NPCs go, Hope in XIII-2 was far better than anything XIII had to offer (I can only remember Jihil and Cid in that game and both of them were horribly under developed). Yeah, Snow and Alyssa suck, and Yuel is kinda weird, but some of the people you interact with in the Fragment quests were kind of cool and there was none of that in FFXIII. The main thing that the original did better was that it had more Lightning and less Serah.

The Story
FFXIII-2 had a bad story. Bad for even a time travel story. And the ending did it no favors. I didn't think FFXIII's story was all that great either, and it's ending had some bad moments, but it wasn't the train wreck money-grubbing attempt that this one was, so I will have to give it the edge here.

Experience System
I thought FFXIII-2's Crystarium was worse than XIII at first, but it grew on me and now I like it better. It's also easier to max out your characters, which I appreciate since grindfests are dumb. The monster aspect also adds a nice variety, and it is a surprisingly deep system. One thing that bothered me was that the abilities for each role were in a set order while the original allowed a bit more flexibility if you were gunning for one particular ability.

Sidequests
Another default win for XIII-2, since FFXIII had only one sidequest and it was boring thoughtless monster hunts (I guess farming gil for hours to upgrade weapons counts too). XIII-2 has all the Fragments (which require different things), a casino level with Chocobo races, monster collection and enhancement, secret worlds, multiple endings to find, and tons of hidden treasure.

General Gameplay
I feel as though the goal of FFXIII was to get to the end, to experience the story. Problem is, the story wasn't that great. Certainly not on the same level as other 'interactive movie' games such as the Uncharted series. I feel FFXIII-2 was different. The goal was not simply to get to the end, the goal was to play the game. You could go in different directions if you so chose. You can go back and replay areas, find new worlds with separate storylines, work on your monster collection, etc. It felt much more like the older FF games where you are embarking on your own adventure, rather than someone else's carefully scripted one. I really liked playing the game, and even having beat it I want to go back and find what I missed (which I had no such desire to do in FFXIII).

So yeah, other than story, and possibly characters (though I say that one is even), FFXIII-2 beats the original in every way. I hope SE builds on what worked in this game (freedom, sidequests, more whimsical), and improves upon what didn't (story, DLC-baiting).