It really varies from shop to shop and even project to project as shops refine their own particular job descriptions and organization.

In general if there is a game director listed this position would be involved with creative decisions and communicating them to the team while the game producer would be more involved with schedule and business decisions and dealing with investors / publishers. In some shops what they call a producer is also driving creative direction and works close with the discipline leads on creative decisions much like a game director. In other shops the game design lead is effectively the game director.

Sometimes the design lead and producer attempt to split the duties of a game director, often with mixed results.