The word "Desire", while it could be used without this connotation, is often used to communicate that you have a need for, or a strong wish or want for something. It is often used in a sexual, a deep longing, or a lustful sort of context, as in: "He desired her".
The word "Wish" can be more widely applied without worry of a miscommunication. "I wish I could go to the moon", "I wish my roommate would do her dishes", and "I wish my friend would answer her phone so we could talk" are all perfectly great uses for this verb.
Basically, the word desire sounds out of place a lot more often. Changing the verb in the sentences above gives you:
"I desire that I could go to the moon" although it would probably be understood, sounds wonky and odd.
Saying: "I desire my roommate would do her dishes" could very much send the wrong meaning to someone listening, as they may hear "I desire my roommate" and then immediately stop listening, since this sounds like you're implying a sexual wish. It just doesn't fit. Same with the 3rd sentence about your friend.
"Wish" is definitely the safer pick in the majority of contexts.