Weary means "tired, fatigued, physically or mentally exhausted from hard work."
Wary means "watchful, cautious, being on guard against danger."

Few people write wary when they mean weary, but the incorrect use of weary when wary is the correct word is quite common.  People who make this mistake are perhaps unaware that the word wary even exists, since weary and wary sound very much alike.  However, wary has the same roots as aware and beware.  It has nothing to do with tiredness (weariness) or fatigue but refers to watchfulness (wariness) or caution.

The adverb form of wary is warily.  "He eyed the snake warily" means that he viewed the snake cautiously, watchful for the possibility that the snake might strike.  "He eyed the snake wearily" means that viewing the snake fatigued him, that he was probably bored with watching the snake.

A student once wrote that he was "becoming weary of spelling errors."  What he intended to say was that he was becoming watchful for such errors and trying to avoid them:  he was becoming wary of spelling errors.   My comment:  "No, you are becoming wary of them.  I am becoming weary of them."