They look fairly close and there is some overlap. There are some smiles I could classify as possibly both a sly smile and a smirk. However there is a difference. A sly smile would imply that I know something you don't; that I am hiding something from you and am proud of it. A smirk is more obvious. I'm not hiding anything. It is a smug look of superiority, and it says that I either don't care about others or think less of them; that I have won in some negative way.

Sly is an adjective, meaning marked by skills in deception. Yesterday he thought he was sly while cheating on his math test. Smirk can be a noun or verb. Ernest, why do you have a smirk on your face? Are you planning on doing something bad? I can tell she is up to no good; she's smirking right now!