You can use weary in quite a few ways, I'll give you a couple of sentences. "I've grown weary of you two fighting all the time!" <---- In this case, weary is being substituted for 'tired' to say that 'I'm tired of this', however in this case, weary is much more extreme. "The man gave a long, weary sigh" <---- Also to signal tired, but it's far worse than being tired. You're exhausted, you're VERY tired and you just can't do it anymore.