You look weary = You look tired. Weary is an adjective; wearily is an adverb. "Anna looked wearily at the hill in front of her. She had been hiking for hours and couldn't wait to make it back to the campground." "Jason, I can see you didn't get enough sleep last night. You look weary!"

You look wearily. - If the quote was "You look weary" then yes, the definition would be "You look tired." But "You look wearily" means that you are gazing (out a window, or at the ocean) in a tired manner. You could also say it like "you tiredly look" - that would have the same meaning, but it sounds weird.