The word weary means 'tired, fatigued'.
The word wary means 'cautious, alert to danger'.
They are not synonyms!
If you're wary of something, you've become suspicious of it; you suspect it might not be all that it seems and it might be dangerous.
If you're weary of something, you've become fed up with it; it's tiring you out and now you're exhausted,either physically or mentally.
Sure, sometimes you can use both words in the same context. If someone is always lying to you, and you've got tired of it, you can say both "I'm wary of listening to him" (because he always lies) and "I'm weary of listening to him" (because he always lies). But those two sentences each mean something different!
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