Desire is something you really want. Like a craving. You are very tired so you desire sleep. You are very hungry so you desire food. You desire to go to Japan. Wish is something you want to do or have but can't at the moment. You are hungry and you wish you could eat but you don't have food. You wish you could buy food but you don't have money. Wish can also be used for simply "want". For example you go to a restaurant and you say "I wish to sit at a table" it means that you would prefer and want to sit at a table