What is Someone who studies art?
A person who studies art in the most broad sense is an
aesthetician. The study of the principles and definition of art is
the field of aesthetics. Sometimes, this word is used to include
the study of all sensory experience, as derived from nature as well
as art.
One can say that an art student is "one who studies art" as
well, but I think you mean "someone who studies art
professionally".
This does not mean that an aesthetician cannot also be an artist
or that an artist cannot also be an aesthetician. Ayn Rand is a
good example of person who was an artist and also an aesthetician.
She wrote fiction literature, but also proposed and analyzed
aesthetic theories.
This is not to be confused with someone who studies the history
of art, which is an Art Historian.