More or less those two words express very similar, possibly identical ideas, they seem to be closely related synonyms. I tried to define a rule and differentiate between those two terms but my attempt was not sucessful. I am also not a native english speaker. On the other hand there are some common phrase examples which prove the unique identity and syntax of each of the two words. In the following sentences: " Ibrahim has no hope to win the first prize in chess " / "I hope you don't mind me borrowing your calculator for a moment, do you? " " Make a wish " / " I am so glad that her wish came true " the word 'hope' can't be replaced by the word 'wish'. As I perceive the meaning of those two nouns is that 'hope' expresses mostly a personal ambition towards a desired outcome where 'wish' is almost like a prayer for a certain event or a desired outcome; but my opinion in this case lies far from providing a proper definition. In the case of the verbs 'hope' and 'wish' their meanings seem to me even more closely related

I'm not a native English speaker, I'm from Spain, but I think I can help with this. When you use 'hope', you're talking about something you want to happen, and you have some expectations it could happen. At least, there is a reasonable possibility for it to happen the way you want. But when you use 'wish' you are simply stating something you like, without any clue if you find it likely or not. For example, after an exam, you could hope you have passed it, in the sense that you have studied and worked hard, and you think there are possibilities for you to pass it. But when expecting to win a big prize playing lottery, you would use 'wish', because it's just a desire, not based in any facts.