EDIT: watched the video now. Jumping looked dumb. Battles are basically the same as twelves but they go to separate screen and appear to have a lot more menus!? animation looked stiff. The battle music sounded like music maybe suited for a trailer or the little preview movie that happens if you dont press start on the main screen for 30 seconds.
Oh well, it happens.
Actually, battles seem to take place in the same field, there's just that quick transition to position the characters on the field. You can only control one character in the demo but apparently the demo is a build from 2008 and the finished version is going to be different but how different, I have no clue. FF13 gets rid of magic points and such and uses time as battle currency for different attacks, magic and so on. You can execute a bunch of smaller attacks or wait until the bar is full enough to execute a bigger one, or then you can just execute one or two commands and not wait until the bar fills till the end. The point is to chain attacks with your party members for bigger bonuses and more damage. I think the battle system seems really cool and fast-paced, I just hope that you can actually control your other characters (at least to some extent) and it's not just some bullshit Square made up.
I watched the videos on IGN and a bit of gameplay video on youtube and I think it looks good. The in-game graphics seem to be better than I thought they would be and the cutscenes don't seem nearly as stiff and unrealistic as in almost all RPG's. I'm a bit disappointed by the battle transition though since I thought the battle would have started right when you run into an enemy but at least the battle transitions are really fast. The music definitely didn't impress me though and I doubt the soundtrack is going to be any special. The whole demo was really linear like FF10 and I really hope the whole game won't be like that. It doesn't have to be as open-ended as FF12 (by jrpg standards) but I want some amount of exploring.
But overall I think FF13 seems promising, probably the only RPG I have really looked forward to since Suikoden 5 or something. I don't have an opinion about the game's story yet though since there's so little we know. I do dig the mix of scifi and fantasy though. And the main characters (assuming they are Lightning and Snow) seem much better than Tidus, Vaan or Zidane so that's a plus.