Could you tell me the difference between American and British drama therapy?

There are some real differences in practice and theory as well as many commonalities. One difference is in the spelling: Drama Therapy vs. Dramatherapy. I encourage you to do some reading to find out for yourself. Try visiting www.nadt.org and look at their suggested reading list. Some prominent British authors are Sue Jennings, Alida Gersie, Phil Jones, and Roger Grainger.