You use asleep and sleeping to describe people when they sleep.
You only use asleep after a link verb like ‘be’ or in the phrase ‘to fall asleep’.
1. Don’t make any noise, Guy’s asleep.
2. It is good, too, for babies to get used to falling asleep in their own bed.
When you use sleeping as an adjective, it comes in front of a noun.
3. The house was full of sleeping children.
4. Ginny stood looking down at her sleeping mother.