No they are not the same. "I had a good time watching the movie" implies that you had a good time while you watched a movie but does not reflect how you felt about the movie. "I enjoyed the movie" shows your impression of the movie you watched.

On a grammatical level, "to have a good time" is complete. "Enjoy" is incomplete because it is transitive: you must tell us what is enjoyed. As Jeff pointed out. "I had a good time watching the movie" may not be about the movie itself; perhaps the company was good, or you were already in the mood to enjoy the movie.