What are the two types of gymnastics most commonly seen in competition and what is the difference between them?

Im not fully sure what you mean. Here are some possible answers:

• Women's and men's. The difference here is obviously the gender, as well as the apparatus they perform on.
• Traditional gymnastics and Rythmic gymnastics. Rythmic gymnasts don't use bar beam or vault, they perform highly choreographed routines on floor, using props such as a ball, hoop, or rope.
• Artistic gymnastics or power gymnastics: Artistic gymnasts incorporate more graceful dance elements into their routines