Is karate better than gymnastics

Better for what?

Self-defense - Karate is certainly better

Competitions - Depends on the style of karate, some Martial Arts have lots of competitions. If you don't like to compete, find a style that doesn't focus on sport. Gymnastics is much more competitive and is recognized globally. You can compete in the Olympics in Gymnastics, you can't in karate.

Longevity - Most gymnasts don't compete after they hit their early 20's. Then what are you going to do? Many martial artists are still actively participating into their 70's and 80's.

Traditional karate does not require any equipment and can be practiced in a relatively small area. Gymnastics use a variety of equipment types, most of which are not available in the home.

Cost wise, it will depend more on the program and training school as to whether there is any advantage.