If you google "hope vs. wish", you will find many webpages that discuss this issue. My own view is that "hope" is used when the person believes that the thing desired could actually happen in the future, or could have actually happened in the past. The verb "wish" is used when the person believes that what he desires cannot happen, is very unlikely to happen, or didn't happen in the past.
Examples of "hope": "I hope he will visit me tomorrow." (I'm not sure if he will visit me, but there is a real possibility he will visit me.)
"I hope he got a good grade in his math test yesterday." (I'm not sure if he got a good grade, but there is a real possibility that he got a good grade.)
Examples of "wish": (Today is Sunday.) "I wish today were Friday." (The thing desired is contrary to fact.)
(Mary is a very unfriendly person, and you are sure she won't help you.) "I wish Mary would help me with my homework." (You use "wish" because you believe that what you desire is unlikely to happen.)
(John died last week.) "I wish John hadn't died." (The thing desired is contrary to what happened in the past. That is, it is contrary to reality.)