The meanings overlap. "Hope" suggests something you expect, something possible. "Wish" just suggests a feeling of longing, of desire, sometimes for something you think is impossible.
"I hope X" means "X is possible. In fact, I think X is likely. I want X to happen. But I am anxious. X might not happen."
X can be something trivial. On the day of a camping trip, "Gee, those are dark clouds. I hope it won't rain."
X can be something profound. "Grandfather was taken to the hospital today. He's very sick. I'm worried." "Don't give up hope. There is always hope."
"I still hope for world peace."
"Wish" means something hypothetical. It is something that you picture in your mind. It's not true now. You would like to "get your wish." You would like your wish to "come true." It carries a feeling of asking the universe for something.
It is a very hot day. You say, "I wish I had a glass of ice water right now." You don't really expect this wish to come true. But, maybe you turn the corner and it is in Wall, South Dakota and you see a sign saying "Wall Drug: Free Ice Water." Your wish came true.
You are a child in the US and it is your birthday and you are having your birthday party. Your parents bring in a cake with candles on it. You are supposed to blow them all out in one breath, and then make a wish. You are supposed to keep your wish secret. You think to yourself "I wish I had a puppy." How likely is it that you will get your wish?
"I wish I had wings like a bird and could fly."
In the United States when the planet Venus is the "evening star," it can be a "wishing star." You might know the scene in Disney's "Pinocchio" when the star shines through Geppetto's window. You might say, out loud:
"Star light, star bright,
First star I see tonight:
I wish I may, I wish I might
Have the wish I wish tonight."
Then you make a wish, but--as with the birthday party--you keep the wish secret.