drow·sy
/ˈdrouzē/
Adjective
Sleepy and lethargic; half asleep.

sleep·y
/ˈslēpē/
Adjective
Needing or ready for sleep.

I think the real difference for you is that "drowsy" is an established term. When you read the list of side effects of most medications, you would most likely happen upon a mention of "drowsy", but not "sleepy".

In other words, "sleepy" is in Plain English.
"drowsy" is a medical term.

Other than that, no real difference.