To clarify, since I've been playing(a little) MP:
You gain XP from fights in story mode fights, VS CPU, and VS Ad-Hoc. You get the XP on whatever character you used at the time. Even if you lose, you'll still get some XP. If you win, you get more XP, as well as Gil, and 'BP'. Gil is used to buy equipment items, and BP is used to buy all the neat little extras in the game, such as extra characters, new attacks and such, as well as more aesthetic things like display pictures and what-not.
Higher level means all your stats get raised, akin to a typical RPG, as well you start out with more brave points. In VS Ad-Hoc, the higher leveled opponent can handicap themselves if they wish, but this is optional. In VS CPU, you can make the CPU up to your level, as well as assign different difficulties. You don't lose your equipment if you handicap yourself though, so you'll still have a slight advantage over your opponent if you have up-to-date equipment on. Which I don't bother with unless I'm in story mode.