Weary usually means emotionally tired. I felt weary after crying for two days. It is not used in the U.S. as much as tired or exhausted. Tired is when you have not slept enough, or when you are annoyed with something. I am tired because I only got four hours of sleep. Or, I am tired of all his lies. Exhausted is a higher degree of being tired. I am exhausted after running a marathon. Or, I am exhausted because I haven't slept in two days.