I hope that you will enjoy your stay in Japan when you visit me next year.

However, 'wish' is a little stronger, but can not be inserted as a direct replacement for 'hope', without a change to the grammatical construction of the sentence. For instance:

It is my wish that you enjoy your stay in Japan when you visit me next year.

Now it sounds like a bullying command and could almost be interpreted as:

YOU WILL ENJOY YOURSELVES, BECAUSE IF YOU DO NOT YOU WILL BE IN BIG, BIG, TROUBLE WITH ME!

'Wish' is so personal that it's best used for things, you yourself wish for, rather than wish something for others, so the second sentence doesn't sound right. You can still wish for others as in: wishing for world peace, that's a wish for others, and is okay. 'Hoping' for world peace sounds a little weaker.

If someone annoys you, they will be in doubt to your growing anger if you say, "I wish you would be quiet'.

Saying, "I hope you will be quiet" means the person might or might not obey you. It's not as strong.

Imagine if you are invited to a meal and are told, "It is my wish that you enjoy this meal I have prepared for you". Put like that, you'd be too frightened to ask the host for salt or pepper to go with the meal for fear of upsetting the host! If on the other hand they merely 'hoped' you enjoyed the meal, you'd feel the host was keen to please you, and you would feel much more comfortable.

'Wish' is strong, 'Hope' is more polite.

Using 'hope' as in the first sentence is the better of the two.