small livestock vs farm animal

small livestock and farm animal both are nouns.

Word	Noun	Adjective	Verb	Adverb
small livestock	Yes	No	No	No
farm animal	Yes	No	No	No

As nouns, farm animal is a hypernym of small livestock; that is, farm animal is a word with a broader meaning than small livestock:

small livestock: small domesticated animals (chickens, geese, rabbits, hogs, ducks, pigeons, etc.) kept for use or profit.
farm animal: any animals kept for use or profit

Other hypernyms of small livestock include livestock, stock.

small livestock (noun)	farm animal (noun)
small domesticated animals (chickens, geese, rabbits, hogs, ducks, pigeons, etc.) kept for use or profit.	any animals kept for use or profit