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 expression like the snarl on "so brauch ich Gewalt"; (viii) the different choices Schubert made in this composition compared with Carl Loewe's setting of the same text.

(I recognize that this doesn't explain why I appreciate these aspects of the performance. But I think it's still useful to give an indication of how complex the phenomenon of music appreciation is.)