I had to look this up recently for a friend, and BLOGGHOREA has answered the question far better than I could have. See below:

“…it is also the Brits who took it upon themselves to add a superfluous “e” to the end of “dramaturg,” thereby not only recuperating the term as British (or at least as less Teutonic), altering its very pronunciation whilst at it.”

– Mead Hunter, “Dramaturg vs. dramaturge,” Blogghorea

To summarize, dramaturg=American English, dramaturge=British.