"Sleep" is the main verb, meaning the state of rest you experience at night in bed.
To "fall asleep" means to enter into that state of sleep! If you are tired and your eyes begin closing by themselves, you could say "I'm falling asleep, I need to find a bed".
Here is an example that uses both verbs, to show the difference:
"Every night I fall asleep at 10 o'clock. I sleep until 8 o'clock in the morning."
In the example, "I fall asleep" means that I will begin to sleep, and "I sleep" simply means I am in that state of sleeping.
I hope this makes sense! If not, just ask me to clarify more and I will help :)