First, there are two Verben used, "hören" and "anhören". The difference in meaning is rather ephemeral and for most purposes they are identical as @Janka already said.
To explain their difference it is perhaps easier to switch to the activity of seeing instead of hearing because here English has (unlike with hearing) two different adequate words: "to see" and "to watch". "hören" is analogous to "to see" whereas "anhören" is analogous to "to watch".
Having said this:
Both sentences are worded very generally, so for both to be true it is not necessary that you do it all the time, just that it is a typical activity you are undertaking during your free time.
Ich höre in meiner Freizeit Musik.
will include you having i.e. a radio with music running in the background while doing something else. listening to the music is not necessarily your main or your sole activity.
Ich höre (mir) in meiner Freizeit Musik an.
This suggests more strongly that when you listen to certain music pieces this will be your main activity. This effect is even stronger when you include the "mir" and use "anhören" as a reflexives Verb (reflexive verb).