Should the characters have different abilities?

No thanks, that would cause too much non-sense with picking certain characters for certain things. Like if I loved playing a mini-game, but the only way to win mini-game is to pick a character who's good at it like Wario and not Peach (who I use all the time). I wouldn't like if it was something like that.

HOWEVER, if it is like MP7 where everyone gets a special Item or whatever, that's fine, I don't mind each character getting a special item.
Pretty much what everyone above has said, MP is not primarily based on skill sets or character ability but the fact that all the characters represent simple, equal player avatars on the board. While it might be fine and dandy if say for example Toadette excelled at every stat (and at would be a godsend for Toadette fans I imagine :p) and most other characters only good at a few, that would lead to a broken game and probably broken base as well.

The special items in MP7 worked well, and it'd be good to see that idea come back in the future.
I personally think it does depend of how it is. There are some things which I would accept like the orbs, but they should not be too overpowered while some others I would definitely not accept, like if I would always be an huge advantage only because you're using a specific character, I don't think it would be good, if they have more power or anything like this. I prefer when it's fair in that game. Basically like most of you have said.