weary : 侧重指由于连续不断进行某项活动而造成的疲劳，以致感到厌烦。
- I think he's a little weary after his long journey.
- Here, sit down and rest your weary legs.
tired : 普通用词，泛指因工作紧张，劳动过度，休息不足等而引起的疲乏。
- She spoke in a tired voice.
- I was so tired when I got home from work last night that I had a quick nap.