snow leopard vs big cat

snow leopard and big cat both are nouns.

Word	Noun	Adjective	Verb	Adverb
snow leopard	Yes	No	No	No
big cat	Yes	No	No	No

As nouns, big cat is a hypernym of snow leopard; that is, big cat is a word with a broader meaning than snow leopard:

snow leopard: large feline of upland central Asia having long thick whitish fur
big cat: any of several large cats typically able to roar and living in the wild

Other hypernyms of snow leopard include cat.

snow leopard (noun)	big cat (noun)
large feline of upland central Asia having long thick whitish fur	any of several large cats typically able to roar and living in the wild