Wii Fit Plus has 15 new games, more yoga, more strength training and is easier to use. It also has a multi player mode. It's way better!!!!! I love the new games, especially the Segway one.
I researched both before asking for the Wii Fit Plus for Christmas. I haven't tried it yet because I just got home (which is where the Wii is ) but I read that the big difference is that there's more stopping in between exercises/games with the wii fit. The Wii Fit Plus is supposed to allow you to put together a workout routine, and keeps you going the whole time instead of stopping.
If you have the Wii Fit already, then you just need the Wii Fit plus disc and it will transfer all your data to Wii Fit plus. But if you don't have a Wii fit board yet, then the only version you can get is the Wii Fit Plus I believe (I think it's the only one they sell with the board now).
Wii Fit and Wii Fit Plus are both games and are sold separately (~$20). In order to use either, you need the Wii Fit Balance Board, which can be purchased as a bundle with Wii Fit/Wii Fit Plus(~$90).