desire vs wish

desire and wish both are nouns.

Word	Noun	Adjective	Verb	Adverb
desire	Yes	No	Yes	No
wish	Yes	No	Yes	No

As nouns, wish is a hyponym of desire; that is, wish is a word with a more specific, narrower meaning than desire:

As verbs, wish is a hyponym of desire; that is, wish is a word with a more specific, narrower meaning than desire:

desire (noun)	wish (noun)
an inclination to want things	the particular preference that you have
the feeling that accompanies an unsatisfied state	an expression of some desire or inclination
something that is desired	(usually plural) a polite expression of desire for someone's welfare
	a specific feeling of desire

desire (verb)	wish (verb)
feel or have a desire for; want strongly	invoke upon
express a desire for	order politely; express a wish for
expect and wish	make or express a wish
	feel or express a desire or hope concerning the future or fortune of
	to hope, to desire or to prefer to have something, or to do something