fool's errand vs mission

fool's errand and mission both are nouns.

Word	Noun	Adjective	Verb	Adverb
fool's errand	Yes	No	No	No
mission	Yes	No	No	No

As nouns, mission is a hypernym of fool's errand; that is, mission is a word with a broader meaning than fool's errand:

Other hypernyms of fool's errand include charge, commission.

fool's errand (noun)	mission (noun)
a fruitless mission	a special assignment that is given to a person or group
	an operation that is assigned by a higher headquarters
	the organized work of a religious missionary
	a group of representatives or delegates
	an organization of missionaries in a foreign land sent to carry on religious work