The ability to have multiple humans together in the Grand Prix, which was not introduced until 64. 64 also brought in a number of new powerups that made you have to attack and defend as well as you raced. I always prefer complicated tracks with multiple routes to them such as Mushroom City or Kalamari Desert over the simple ones.
Don't really care for new characters much since they're all pretty similar and I just stick with Yoshi or Toad most of the time anyway. New powerups are always fun though, unless you're on the receiving end. I noticed that the jumping Feather was only put in the first one, but how about an enhanced version that gives you wings for about five seconds, enough to skip huge parts of some tracks if you use it right (if you don't use it right you'd just fall off the course)?
New (preferably complicated) tracks:
The Warioware Inc. Building
Shy Guy's Toy Box
The Excess Express (as in racing to stay ahead of the train)
Count Bleck's Castle