화가 covers a narrow group of artists. It just means "painter", and doesn't even include sculptors (조각가) or architect (건축가), because 화 indicates a visual representation made on a canvas or electronic screen. 예술가 is a much broader and somewhat loose term. 예술 means "art" in the broad sense, including all fine arts, music and creative writing. 가 denotes a person engaging in a certain trade as a profession, and applies equally to 화가 and 예술가. 아티스트 is fairly close to 예술가 but it tends to mean more modern kinds of 예술가 and performing artists. For example, singers and other performers are often called 아티스트 but not 예술가 so much.

I guess it depends on what you mean by "artist". I know 예술관 means "art gallery", so I think it's safe to assume that an 예술가 has something to do with that kind of art.