The implication is that if you are sad there are more associated tells than just your mouth. Imagine a clown doing a mock sad face and exaggerating sadness by frowning.

I've always felt this was a forced contrivance and never that anyone literally thought frowning had anything to do with smiling.

How interesting - I'm British, and I got the joke immediately. Yet, I'd hardly move my mouth if someone asked me to put on a frown. I'd never really thought about that one before.

Yeah, it's a well known expression, but it only makes literal sense in American English.