The main difference between sleep and asleep is that sleep is a noun and a verb while asleep is an adjective and adverb describing the activity of sleeping. Asleep is an adjective that describes the action of sleeping. Sleep refers to slumber, rest, or a state of inactivity, which is a basic need of a human being.
Sleep denotes to be in a state of inactivity where the body is relaxed and peaceful, and sleep is a word that may also refer to death. It may also refer to inactivity.