Apollo was the god of music because he loved the way the instruments sounded when he played them especially the lyre.
In mythology he doesn't, is this question related to a book or game?
In the beginning it seems he was a god of avenging evil, often getting away with acts either outrageous or severe. A few examples; Killing the Python in the oracle Delphi, a sacred place, and then...
He liked sex a little too much and it often got him into trouble. The same for most gods.
THere is no WHY about the Greek gods' responsibilities. They were born with them.