An example where «Актёр» is possible, and «Артист» is not is the name of the character in Gorky's “The Lower Depths”. The name refers to the past profession of the character who is now a good-for-nothing; what he did in that profession now matters not. You could still say «он плохой артист» about him, certainly, but that doesn't matter in the drama. His role in the drama is not to play tricks before the other characters, so he's not «Артист»; his role is to be a had-been who now does nothing. He is “someone who had a job once”. Which job? Well, an actor.
An example where «артист» is possible and «актёр» is not is a sentence like «Ну, ты артист», when someone makes some bold move or plays an advantageous (for them) trick in the eyes of the spectators. The word refers to the creative character of their trick, to how they form the reality around the others, most likely by deceiving someone or doing something that other people didn't expect, by doing actually one thing yet showing something else to people. But they don't work as actors and aren't even compared to such (the sentence doesn't mean “you're like an actor”, rather it means “you did play it strong”).
I don't think there is any reason to believe the word «артист» in the meaning «художник» (an artist of any kind, or even in the most generic sense of the word before you specify the creative art in question) is obsolete, as someone else suggested. This meaning is, indeed, literary; but so the subject is. It is literary to talk of art in the general sense. When a specific profession is implied, the generic aspects are not so important, and specific aspects are more telling for imagination of the users of the language, so another word that implies these aspects is used («писатель», «поэт», «скульптор», «композитор»).
So, what is the conclusion?
1) When talking about plays or reading novels aloud (like on the radio), my conclusion is this: the denotation of the two words is the same, but they refer to different thoughts: either acting as a job, or creating an impression on the minds of spectators.
2) When not talking about playing dramas or reading novels aloud, «актёр» is out of question. In this case, you would commonly find a singer or a joke-monger (someone who tells jokes from the scene, like Altov or Zadornov) to be called an «артист».
3) Also, «артист» (more commonly, «художник») has the sense “an artist qua artist, an artist in the generic sense of the word”. To express this sense, these two words are the only thing that the modern language has.