big cat vs felid

big cat and felid both are nouns.

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

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

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

Other hypernyms of big cat include feline.

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