I must admit that I don't divide music up - it makes it too complicated.
However, there are such things as listening for education and listening for pleasure - the two aren't necessarily mutually exclusive, as WinterLight points out - but sometimes I enjoy listening for the sake of working out what the musicians are doing, or fitting the music into a historical context, or some other studious reason, and sometimes I enjoy listening just because the music enhances my mood or makes me think of something cool.
There's no particular music that fits either listening criteria, however - I'll happily listen to ABBA and pick the arrangements to pieces, or listen to Penderecki or Messaien just for the spine-chilling thrills of the sounds.
I recently listened to Tubeway Army's "Are Friends Electric" with the express purpose of creating a cover version - purely for fun, but, of course, had to listen to each line carefully to work out what was being played and sung - and tried to trap all the nuances that made it sound the way it did so that I could roughly match the sound engineering.
This is an example of doing both simultaneously - I've got a copy of the recording somewhere...