How does the process of hearing differ between a person who is awake and a person who is asleep?

While you sleep, your brain processes sounds on a basic level, sometimes causing you to wake or jostle. The impact of sound depends on the stage of sleep you are in.

When you are awake, the sound produces some kind of reflex, and therefore the part of your brain that processes the sound kicks in and pulls recollections to see what the sound is.
Additionally, you have got a voluntary reaction. Once your sleeping, there'll still be a reflex, however, there's no process and decision making, thus you'll hear it however not be able to distinguish what it is.