In anime, what's the difference between an art director and an art designer?

An art director tells the art designer to draw the thing the art director wants. The art designer draws it. I can't think of a better explanation.

Director (Kantoku)

The director is responsible for the overall look and feel of the show and is the leader of the production. (Well, this is the idea anyway ) The director determines what sort of show he wants to make and creates a set of storyboards, which are sequential drawings detailing the major scenes of the show, sort of a visual script. They have various information about dialogue, music, camera work, and such and serve as a basis for the animators to create their layouts, then their key drawings from. (Some director's storyboards are more comprehensible than others. Some sketch very rough and some put a lot of detail into the drawings. There is no set way of doing it other than that everyone uses similar paper forms to work on.)

Enshutsu

Often translated as animation director or technical director. Enshutsu is one of the most difficult and most important jobs in the Japanese animation industry. The enshutsu is between the director (kantoku) and the production staff. He is responsible for checking and supervising the show through the production, from initial story to the final released product, and in many cases, has almost total control over it. He checks the animation drawings as they are being done, sets up the scenes before they go to camera and supervises the sound and voice recordings and all the editing amongst many other jobs. The exact job and responsibilities vary from company to company, and from show to show as well. Sometimes the enshutsu ends up as the whipping boy for the director; sometimes he or she carries the show and the director sits back and watches. On larger productions there is sometimes more than one enshutsu and usually quite a few assistants. It is important to have a good knowledge of animation production as well as artistic talent to do the job and it usually takes four or more years in the industry before someone can do the job right.

Character Designer

As the title states, this person designs the characters for the show. Although this may appear to be a very creative and interesting job at first glance, it is a very demanding and difficult job as well. Some designers are given great freedom to design what they want to but more often they have to work within the constraints set by the directors, original creators, producers, sponsors or other parties. The character designer creates a settei or model pack which contains each of the characters and defines the costumes they wear and what they might carry. The character designer should provide as complete a model as possible for the animators to work from or it becomes easy to go off model and makes the job of the animation supervisor (see below) even harder.

*Note: In the anime industry, there are various Art Directors and various Art Designers, so it depends on which ones you want to know about. Generally, the difference between them is that one of them directs/supervises artwork, while the latter designs things to the request of the director.