Or like "I listen to everything" and "I'll listen to anything."

To me one says I love every song and the other is more, I'll listen to what's ever on the radio and give it a fair chance.

Absolutely. I know a lot of people who listen to music that they don't actually enjoy! Listening to a certain type of music doesn't necessarily mean you like it, but some people might sometimes give something another chance if they think it'll grow on them.

Or, when going by your second set of quotes ("I listen to everything" and "I'll listen to anything"), the former is saying that they have a wide variety of music that they actively listen to, whereas the latter is suggesting that they are willing to venture outside of their usual taste (that is, if they have a usual taste) and can give any kind of music a chance. They may or may not end up enjoying the music.