We use the Present Simple to talk about regularly scheduled future events. As the film likely starts at 9pm every night, the Present Simple is the correct choice.

Both are grammatically correct. Both mean almost exactly the same thing. In fact we can add two more to the list: The film is starting tomorrow at 9 pm. The film starts tomorrow at 9 pm. The film will start tomorrow at 9 pm. The film will be starting tomorrow at 9 pm. These have very slight differences in meaning. "The film starts tomorrow at 9 pm" is good because it is short and simple. Both "is starting" and "starts" are interesting, because they are using what is grammatically the present tense to talk about the future. This is very common. We put our minds into the future, and talk about the future as if we were already in it. The following may help understand why there's more than one way of saying the same thing. "The film starts tomorrow at 9 pm" suggest that's the moment when the room darkens and opening credits begin. Now imagine a weekend of activities, perhaps at a conference. There are talks, lunch, dinner, a film, drinks after the film. You might hear that and say "My plane is arriving at 6 pm, can you tell me when the film is starting?" "The film is starting at 9 pm."

IELTS with Joe gave an excellent answer. If you have the excellent book English Grammar in Use, consult unit 19.

"Starts tomorrow" is correct, because it's for indicating a general information or fact (present tense)

Starts