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Both have same sounds but different meaning.
Wary – Being careful about possible dangers or problems
Example: Ada’s dog that has been maltreated often remain very wary of strangers. (Another word to replace wary are Careful, alert, cautious)
Weary – feeling tiredness due to lack of sleep or as a result of excessive exertion
Example: He gave a long, weary sigh (Another word to replace Weary are “Exhausted, tired, worn out, drained etc..)