What is the difference between these two words? I’m having some trouble figuring it out.
D’oh.
One’s a verb or noun and the other’s an adjective???
Am I missing something???
I need to sleep. (verb)
I need some sleep. (noun)
I’m asleep. (adj.)
simple 
So…
The baby is sleeping. (Verb)
The baby is asleep. (adj)
Is it just that simple?
yup.
Yeah, it’s just a weird combination.
Especially if you factor in usages like “falling asleeep” and “waking up”.
ADJECTIVE: 1. In a state of sleep; sleeping. 2a. Inactive; dormant. b. Indifferent: politicians who are asleep to the needs of their constituents. 3. Numb, especially from reduced circulation of blood to a limb: My leg is asleep. 4. Dead.
ADVERB: 1. In or into a state of sleep. 2. In or into a state of apathy or indifference. 3. Into a state of numbness. 4. Into the sleep of the dead.
I had asleep while she showered. 
I’m an attorney and I handled asleep and fall case for her after she injured herself in the shower.
Sleep is what you try to do when you’re not at work. Asleep is a huge bookstore on Dun Hua S. Rd. just north of Ren-Ai Circle.
See? That was easy!
Asleep is stative (a state). Sleeping is dynamic (a process). I think that’s what a semanticist would say.