It's a hierarchical difference. Creative director > art director > designer.
Or you could think of it this way: an art director has to manage multiple designers. Consider a large project where the work can't be performed by a single person. In that case an art director is responsible for taking the best ideas of all the designers and integrating them into a cohesive vision.
Creative directors handle multiple art directors. They operate more at an agency level.
For example, in the movie poster industry the creative directors are like partners at a law firm. They decide on the direction the entire agency is going to take, what kinds of movies or video games they want to compete for, etc. The art directors are assigned to manage projects using the designers.
You need two kinds of skills--people skills and design skills. Ideally, you should be a good enough designer that you can objectively evaluate the strengths and weaknesses of any design and come up with a workflow for solving problems. ...And you have to be able to manage creative people, which is a difficult thing to do.