They have a very similar meaning!
"Get dressed" is a colloquial, more common way to say "put on your clothes.
It comes from "get dressed up" - which is what people do when they're going out to a fancy event - like a dance, a fancy dinner, a party, etc. Now people just say "I'm getting dressed" in a variety of situations - like when you're getting ready for your day. "Put your clothes on" sounds more stiff and formal.