These are two distinct words. They aren’t quite homophones (there and their are homophones, two distinct words that have the same pronunciation)2 but they do sound sufficiently similar to create confusion.
To be wary is to be cautious or concerned “about possible dangers…”.1 E.g., “When that large barking dog jumped over the fence I became wary of continuing in that direction.”
Weary is a quite different word and idea. To be weary is to become tired or perhaps bored as in “I weary of this long speech.” To be wearying is to be exhausting or perhaps boring.
Be wary of confusing wary with weary or you may quickly become wearying.
NOTES
1. I wanted to use the word homophone in honor of this bizarre story.
2. s.v., “Wary,” New Oxford American Dictionary.
I have heard documentaries on PBS where the narrator did not know the difference between secession and succession! How about too and to? I could go on, but….
I looked up the bizarre story. I would imagine most languages would have homophones, most exceptions being in the tonal languages, but I don’t know whether even THEY would be exceptions. Modern Greek (& Byzantine-pronounced Classical/Koine Greek) has plenty..