wild-goose chase vs pursuit

wild-goose chase and pursuit both are nouns.

Word	Noun	Adjective	Verb	Adverb
wild-goose chase	Yes	No	No	No
pursuit	Yes	No	No	No

As nouns, pursuit is a hypernym of wild-goose chase; that is, pursuit is a word with a broader meaning than wild-goose chase:

Other hypernyms of wild-goose chase include pursuance, quest.

wild-goose chase (noun)	pursuit (noun)
the fruitless pursuit of something unattainable	the act of pursuing in an effort to overtake or capture
	a diversion that occupies one's time and thoughts (usually pleasantly)
	an auxiliary activity
	a search for an alternative that meets cognitive criteria