This is an excellent question. grouchymusicologist above has it right that "music enjoyment is a remarkably multifaceted phenomenon", and I would like to expand on this. Michael J. Parsons, in How we understand art: a cognitive developmental account of aesthetic experience, identifies a sequence of developmental stages in the appreciation of visual art. This is of necessity a very rough and un-nuanced summary since I don't have the book to hand, but I think this sequence is: first, colour ("this painting is red"); second, subject matter ("this painting is of a dog"); third, emotional content ("this painting makes me feel wistful"); fourth, technique ("this painting is pointillist"); and fifth, historical relationships ("this painting is a witty riposte to a work of Velasquez"). I can't point you at a corresponding developmental study of music, but I'm sure that similar stages of appreciation are there. To give a flavour of the different kinds of thing going on in the appreciation of music, let's take an example: here's Ian Bostridge singing Schubert's setting of "Der Erlkönig" by Goethe. When listening to this, I appreciate: (i) the timbre of the piano and voice; (ii) the driving and urgent rhythm; (iii) the words and the story; (iv) the way the harmony creates and releases the dramatic tension at appropriate points in the text; (v) the skill of the performers: stamina is needed by the pianist to keep the triplets going, and vocal control by the singer to maintain timbre of the high notes; (vi) the "tone-painting": that is, the ways in which the musical notes illustrate aspects of the story, for example the repeated notes representing the horse's hooves; the way that the "child's" entries are a semitone above the piano, this discord illustrating his distress; the way that each entry is higher and more distressed than the previous one; (vii) the vocal acting of Ian Bostridge: his use of different vocal timbres to differentiate the four parts, and details of expres

I like to draw a (rather pretentious) delineation between music and songs, the archetypal examples being, say Beethoven's fifth symphony and "Call Me Maybe". (As a side note, I very much consider it possible for something to both be a "song" and "music") I enjoy music because I played a few instruments and sang when I was younger, so I know enough musical theory to appreciate the artistry it took to come up with the structure in the music, and (when appropriate) lyrics. Contrarily, I enjoy songs (although happen to hate "Call Me Maybe") because they're fun and upbeat and keep me in a positive mood. You can find a song to fit most moods, and fitting them very closely is a very satisfying feeling. Once last week, I was in a very relaxed mood on my way home and set Sultans of Swing on repeat, because it fit how I was feeling very exactly. It was probably the happiest I have been in the last week. Additionally, sometimes songs have the ability to change my mood and/or motivate me to work harder, and I often exploit them for this purpose.