Painter: someone who is primarily interested in getting various effects from the physical qualities of paint.
Designer: Harder for me to put this in a sentence: someone who plans out part or all of the form of an art work or some other object.
I'm aware that a painter is a kind of designer, but here I think the difference is that a designer would be less concerned about the rendering of their drawing/painting and more concerned with the execution of the design, the final image or whatever that their drawing is a plan for, which could be a muppet or a video game.
I think the distinction is important because it determines the way you go about your work. For example I consider digital "painting" to be a form of video, anyway this gives rise to different "what if" scenarios than if I were to believe that I was a painter, basically a different set of rules. What do you think?