Wary: to be cautious and aware, on guard
I am wary of links in emails from people I don’t know.

Weary: exhausted or very tired. It can be physically tired or an exhaustion of energy, tolerance and spirit.
The weary CFA volunteers were grateful for a cold drink.

Thinking “if you were in a war, you would need to be wary of your environment” may help you tell these two definitions apart.