felid vs big cat

felid and big cat both are nouns.

Word	Noun	Adjective	Verb	Adverb
felid	Yes	No	No	No
big cat	Yes	No	No	No

As nouns, big cat is a hyponym of felid; that is, big cat is a word with a more specific, narrower meaning than felid:

felid: any of various lithe-bodied roundheaded fissiped mammals, many with retractile claws
big cat: any of several large cats typically able to roar and living in the wild

Other hyponyms of felid include cat, true cat, cat.

felid (noun)	big cat (noun)
any of various lithe-bodied roundheaded fissiped mammals, many with retractile claws	any of several large cats typically able to roar and living in the wild