saber-toothed tiger vs big cat

saber-toothed tiger and big cat both are nouns.

Word	Noun	Adjective	Verb	Adverb
saber-toothed tiger	Yes	No	No	No
big cat	Yes	No	No	No

As nouns, big cat is a hypernym of saber-toothed tiger; that is, big cat is a word with a broader meaning than saber-toothed tiger:

saber-toothed tiger: any of many extinct cats of the Old and New Worlds having long swordlike upper canine teeth; from the Oligocene through the Pleistocene
big cat: any of several large cats typically able to roar and living in the wild

Other hypernyms of saber-toothed tiger include cat.

saber-toothed tiger (noun)	big cat (noun)
any of many extinct cats of the Old and New Worlds having long swordlike upper canine teeth; from the Oligocene through the Pleistocene	any of several large cats typically able to roar and living in the wild