When I see 聴 my mind basically jumps first to 音楽を聴く. As mentioned this has the meaning of listening to something with some sort of interest in it. You might want to connect it to the word 傾聴, which means "to listen intently," or roughly that. Of course you can use 聞く in that sentence but that kanji has a broader meaning of hearing in general. The difference is basically the same as that between 見る and 観る, as in 犬を見た vs. 映画を観た.
Asking questions has its own dedicated kanji as well: 訊く. You could probably translate this as "inquire," as using that kanji gives it a slightly more refined feel, and I think that this holds true for the others as well. The choice to use these kanji is deliberate and creates extra nuance as to the nature of what you are hearing (vs. listening), seeing (vs. watching), asking (vs. inquiring), etc.
Just a note from the dictionary on the usage of 聴く: