or are they the same thing?

I think you are meaning whats the difference between Hunter and Jumper...
Jumper is trying to go over the jumps in the time allowed and going around corners quickly, hunter is pacing the horse, (I think....) Lol

Jumping and show jumping are the same thing. With show jumping you're just jumping in front of judges.

Show hunter and show jumper are very different though. With show hunters, you aren't trying to race through the course. You go through it at a much slower pace, making yourself and your horse look good- which is what hunter classes are all about. Show jumping is trying to complete a course in the time alotted with the least amount of faults, and judges don't judge your riding (as in your form and your horse's form).

It's the same thing. A jumper horse is competeing in show jumping. Usually people say "I own a hunter" or "I own a jumper". Most of the time people just say hunter/jumper which basically means anything related to jumping.