When you hope for something you believe there is at least some likelihood it will happen.
- I hope it stops raining soon.
- I hope I will pass my test.
You can use it to express regret:-
- I had hoped it would stop raining.
When you really don't think there's much chance of something happening, you can use wish:-
- I wish it would stop raining.
You can use wish to express regret:-
- I wish it had stopped raining.
- I wish I had paid attention to my English teacher at school.
And of course you can make a wish, as in:-
- I wished upon a star last night.
You can also wish happiness and health and riches for other people.
- "Best wishes."
"We wish you a Merry Christmas."