Would you like listen to music

In short: there is almost no difference when using them in that sentence, and both are grammatical and common.

In general, the "Xing" form is slightly more casual whereas the "to X" form is a bit more abstract. In the form "I like..." it's essentially the same, but the difference becomes a lot clearer when you use it as a subject:

Listening to music is the best hobby

vs.

To listen to music is the best hobby

The first sounds like you are speaking from personal experience, whereas the second sounds more like you are making a philosophical or poetic statement.

Slightly beyond the scope of your question, but one last thing to note is that there are some verbs that require one form or the other, and that will overrule any other consideration:

I enjoy listening to music [correct]

I enjoy to listen to music [wrong]

and

I want to listen to music [correct]

I want listening to music [wrong]