Diddy Kong Racing is different in several ways. First off, there are different vehicles you utilize (hovercraft, airplane, car, and I think the DS version also has the magic carpet). Then, there's also a large environment that you explore to select stages and to find hidden items which help you unlock new stages. Also, each character has its own strengths and weaknesses (as compared to each kart in MK). One other difference is when you're racing, the weapons/tools you collect are not randomly selected (each color balloon you hit in your vehicle has a different power - and they build upon each other, so if you get one red balloon, it's one rocket - if you get two red balloons, it's a homing rocket, etc).
It's a bit more fleshed out of a game, though IIRC not quite as fast-paced and hectic as MK can be.