Traditionally, the producer is the businessman, lampooned in Mel Brooks' "The Producers." The producer collects money from backers, or perhaps a single backer who might in return be titled "executive producer." Tasks such as booking the theater and arranging ticket sales usually fall under the producer's oversight.
The appearance, style and content of the play generally spring from the director's vision. All creative aspects of a production are usually the director's responsibility, though they might be delegated to an artistic director, musical director or heads of departments such as props and costumes.
Although a director might decide on, for instance, a certain style of costuming, the producer is often the logistics coordinator for acquiring elements of the costumes themselves, perhaps contacting a local formalwear company and arranging rentals or cross-promotions. Decision of what script to use can come from either the director or the producer. Casting is another area in which producer and director are typically involved, though here the director is tasked with filling roles, while the producer might deal with things such as artists' contracts.
Not only are there many differences between the roles, there are many differences within the jobs themselves. A director and a producer can each take a large portion of the traditional roles of the other, such as a creative producer who takes care of all aspects of the design of the play, hiring a director to work solely with an actor. Or a director could depend more heavily upon department heads and focus his involvement more on the business end.