Someone who claims to be concerned only with matters of art and beauty is known as an aesthete. Depending on the way he goes about it, he might also be known as a snob.
The Greeks considered an aisthētḗs as "one who feels." It's from this idea of emotional sensitivity that the word aesthete came about in the late 19th Century as a noun for someone who greatly appreciates beauty and the arts. An aesthete doesn’t have to be wealthy or highly educated. You can be a hotel maid who stops cleaning to absorb a strain of Beethoven and be as much an aesthete as the poet who writes about the splendor of a leaf.