A filmmaker is a person who makes films.

A producer makes films.
A director makes films.

So I would say both are filmmakers.

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The aim of an argument or discussion should not be victory, but progress.
Joseph Joubert, essayist (1754-1824)

Both Producer and Director are considered filmmakers.

Producers act more like behind the lines generals, in that they manage the film production, both financially and logistically. They also have a HUGE say in many of the creative decisions that shape the film. Ultimately, it's the producer who is responsible for the final product of the film.

The Director is more like a feild commander, who gets his troops (the cast and crew) to turn out the shots that the editor will need to put together the film.His primary concerns are getting the best performance out of the cast, and getting the kind of angles and coverage the editor will need later in post. The Director's right hand man is the DP, or Director of Photography.

The DP is like the seargent. He actually works with the sound man, the lighting techs, the grips, the set people, etc., etc.. in keeping the production moving. He sets up the shots, gets the scene lit, works with the sound people to make sure the mikes are getting good sound... He also makes the creative choices about the type of filmstock to use, the general lighting scheme, etc.

You might look at the book "The Film Director" by Richard L. Bare. He outlines this stuff really well, and also leads you through the process of directing film.