Design in many fields is a process of creating something functional within an aesthetic guideline.
Art, on the other hand, is the path of taking a creative means to an aesthetic and evocative end.
Both certainly use aesthetics as heavy influences in their end goal, but on a practical level the two are quite different.
A design should be objectively judged against its function. A piece of art should be subjectively judged against one's aesthetic values. In a proper design course in school, one is given hard criteria against which your project is defined for a grade. In a proper art class, you are graded mainly on effort and applied techniques, not on the subjective judgement of the aesthetics of the finished works.
Better put perhaps is to give an example: A beautiful fountain is designed, a classical sculpture is art. If the fountain does not allow water to flow, the fountain ceases to be a fountain. All parameters in creating the fountain must take into account that the end goal is that the fountain shall allow water to flow and all aesthetic considerations must account for the end function of the fountain. A sculpture has no such boundaries, and the only end use is aesthetics alone. Note I have to set aside mechanical and functional sculptures -- these are design not art ;)
You can have an artist create a logo, but it may not, in the end, function as a logo. It may be a very beautiful illustration, but if it does not meet the functional parameters of a logo, it will be an illustration -- a work of art -- and not something identifying a corporate brand.
It is a generally accepted thought that art can be liked or disliked, but is not WRONG. It may be considered skilled, unskilled, "pretty" or "ugly," but it is not judged against a defined set of functional criteria. I don't hold my designs against the criteria of "art". My designs can be wrong, if their form does not facilitate their function. If their form does not facilitate their function, they're unfinished, or need to be re-thought-out.
If you purchase a finished painting, it is bad form to request correction or changes. The art is "done". Design should not be fully purchased until it is done. And for it to be done it must fulfill its functional parameters.
When you purchase a web design or a logo design from Eclectic Tech, you're not purchasing art -- I'm an occasionally inspired artist, but I'm not an exceptionally skilled or talented artist. You're purchasing a design, and I have inspiration, talent and skill in design -- in seeing the functional parameters and applying inspiration towards aesthetics while always testing the overall function against the parameters required.