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.

Very little. One might say that "wish," is more likely to be used for a desire that is unlikely ever to be realised. For example, a tone-deaf person would be likely to say "I wish I could sing so much." But both verbs are used to indicate something that somebody wants. But we don't often use the verb "to desire," in speech. We simply say "I want.........," or "I'd like.............." but if somebody says either I want to I'd like, they are expressing a desire.

Desire sometimes is applied in some place more specific to sex or ambition, so you had better use it with care. Noun: desire - /dɪz.ˈaɪr/ ♫ 1. An inclination to want things. A man of many desires. 2. The feeling that accompanies an unsatisfied state. 3. Something that is desired. Verb: desire - /dɪz.ˈaɪr/ ♫ 1. Feel or have a desire for; want strongly. 2. Express a desire for. 3. Expect and wish. (sage dictionary)