I would say the difference is in the strength of the action. Longing is stronger than desire, and desire is stronger than wish. So I might "wish for a new bike," but not actually expect to get one. If I "desire a new bike," I will probably save my money and go buy one. If I "long for a new bike," it means that I really, really want it. Longing, however, also has a connotation that maybe it's something that you cannot have. For example, I might long to see a relative again who has died, but of course, I cannot.

“Wish” is the most neutral of these words and has the broadest use. “Desire” often has a sexual connotation that “wish” does not have. Example: “In their 50s, the husband and wife no longer felt the same desire for each other than they had in their youth.” “Longing” implies a strong and painful feeling of wanting to have something you don’t have, so it is a much stronger word than “wish”. Example: you might say “I wish that it were sunny today” but you probably would not say “I long for it to be sunny today” unless, perhaps, it had been rainy for so long that you had become seriously depressed.