To be wary is (1) to be on guard against something, or (2) to be watchful or cautious. Weary means physically or mentally fatigued. It’s a synonym of tired.
Examples
Partygoers should be wary of the legal and health implications of using the ecstasy-like designer drug known as “meow meow” despite it not being listed as a controlled drug in Victoria. [The Age]
For the 39th year, the Minneapolis Boat Show is sparking dreams of summer among Minnesotans weary of winter. [KSTP.com]
But one is cautioned to be wary of what one wishes for. [Sports Fan Live]
And though they were weary from their hard work and arduous travels, they were ready to return to their regular service work when they reached New Jersey. [The Times of Trenton]