Words that are close in definition or meaning sometimes confuse me. This is true of wary and leery. They seem very similar. An internet search was not much help.
When you feel or show caution about a possible danger or problem.
Relate words (synonyms) are alert, cautious, careful, chary, circumspect, on your guard or on your toes.
This word usually means cautious or wary (see, it gets tangled here) due to realistic suspicions.
Related words are careful, chary, cautious, distrustful, guarded and suspicious
Both words are adjectives. Usually wary and leery are followed by the word of.
For example, one is leery of strangers.