It does as a matter of fact! To a certain degree. You see, some of the carts have different stats.

[He points to the menu on the side of the screen.]

Some of them are faster, but harder to turn. While others are slower, but turn more easily. But I've played this game so much that I can tell you that it doesn't affect your chances of winning at all. Just pick whichever you like the best, and you'll grow accustomed to the way it controls!