This answer addresses the typical expectations in a corporate type environment. Your specific company could be set up completely differently.
Typically its is about the organization structure. A VP will be the business head of the company and will have several directors and/or managers reporting to them. Their role is typically to set the business direction, help prioritize projects, interact with business partners and customers.
The director role is more of the operations position. The director will have several managers or teams reporting to them. It is the directors responsibility to make sure that business priorities are being advanced and that the resources are in place to meet all of the objectives set out by the CEO, President, and VP. In smaller organizations the Director will also act as manager and will have the additional responsibilities of overseeing and development of their teams. In a larger company the director will not act as a manger, but rather only have the management responsibility over the managers.