While you can google up articles on studies that try and correlate music preference with personality (and there may be validity to these studies) that say that preference for classical is associated with more introversion and interest in complexity as well as aesthetic experience, my own answer runs along these lines:

Larger-scale form, narrative and development is of interest to me, regardless of genre. In rock, a 10-minute piece of music will, other things being equal, be of more interest to me than a 4-minute one. Also, my greatest interest since getting into classical about 3 years ago has been the larger-scale structure of exposition, i.e., symphonic form, as distinct from chamber or solo music. I love symphonies. It may not just be the orchestral "timbre," but just the number and variety of instruments brought to the harmonics. Also, I happen to find many more examples of pieces that invite considerably greater aesthetic engagement in classical than I do in rock/pop. This is not to say that there aren't exceptional cases in the rock/pop world that are of personal importance.

Also, I tend to be drawn more towards music that doesn't feel the need to keep your tempo for you by loudly pumping out a back-beat (or front-beat, if you will). The striking of beats in music tends to irritate me, and sometimes incredibly so.

I'm also not big on the vocal aspects of music, and find it just a bit odd that so much rock/pop feels some urgent need to put words into songs rather than let the music speak for itself.

I can see myself in the not too distant future becoming more interested in jazz, which may require me overlooking the "sparseness" of the small ensembles that are often used in recordings there. But if I get to where I don't mind sparseness, the whole world of classical chamber music might be my next focus.

And yes, I am usually classified as far on the end of the introversion/extroversion scale, highly analytical and critically-minded.