Let me answer in a pretentious film student douche bag voice
A producer is a wide ranging title. It could be someone who is owed a favor by the studio so they threw his name in the credits or it could be someone who oversees the movie from beginning to end attending all the creative meetings, marketing meetings etc.
A director is someone who oversees creative details and physical production of the film big and small.
While the duties of both both the director and producer might overlap for big stuff (costumes, casting, etc) the day to day of production and the technical aspect of the film is mainly the directors job. While bigger business decisions such as corporte sponsorships and contract negotiations with stars fall more into the producers lap.
For man of steel christopher nolan took what they call a "grandfathering" role. So he had a big hands on with the creative aspect of designing the films tone and style, but as far as the final product on the screen im sure its mostly zack snyder.