Listening is the active attempt to hear things.
Hearing happens involuntarily/passively whether you try or not.
You might have difficulty hearing something on the radio, now matter how hard you listen, because you can hear some other noise nearby, people talking loudly beside you.
If you want to hear something soft, quiet, in the distance you must listen carefully.
To add to what Art mentioned, here is my version of the basic differences:
hear --> This is a sense. It’s not active, it’s passive. You hear due to the fact that you have ears.
listen --> This is an active process. You listen because you want to know what is being said, because you enjoy the music, because you want to learn something, etc.
It is possible to hear someone talking without listening to them. The opposite is not possible!
And one more post to address the specific sentences:
You’d normally say “hear someone coughing”.
However a worried mother might “listen to a sick child’s coughing” in order to try to determine whether the cough was getting worse or improving, for example.
You’d normally “hear the doorbell ring”.
When you are paying careful attention to what someone is saying, you’re “listening to them carefully”.
And, one more time, you have to first be able to hear something before you can listen to it.