What are the differences between the two? Why do most people use D-pad instead of analog?

The analog mapping in Spyro 2 and 3 is broken. The angles achieved with down-left and down-right on the dpad are impossible with analog, so you can't turn as sharply.

If you're used to playing with analog, I would recommend switching as soon as possible. The worse you are when you switch, the easier it'll be.