This was a main topic for the analytic aesthetician Monroe Beardsley. In general, the philosophical work done by focusing on the "aesthetic experience" rather than the "aesthetic object" is that it relocates the aesthetic qualities in the experience of the viewer rather than in the object. This move was prefigured by Kant, who did not use the actual term "aesthetic experience," but who considered the key feature of a aesthetic object to be the fact that it inspires/necessitates a "judgement of taste" (which is arguably the same thing, or at least a close cousin).
This opens the door for the resolution of aesthetic paradoxes such as the existence of "found art" --objects that only appear to be art within the proper context--, or the fact that two people can judge the same piece of art very differently. If it is the experience rather than the object that has aesthetic qualities, then these problems are less damning then if the aesthetic qualities are held to physically inhere in the actual art object.
The main difference between the terms "artistic" and "aesthetic" is that the first is a term from natural language, while the second is intended to be a technical term with an exact philosophical definition. In practice, however, neither term is well-defined. Beardsley's own definition holds that an aesthetic experience is the gratification experienced upon correctly and completely experiencing an aesthetic object. However, this definition is arguably circular, or at the least, dependent on some prior definition of aesthetics.