puking vs hematemesis

puking and hematemesis both are nouns.

Word	Noun	Adjective	Verb	Adverb
puking	Yes	No	No	No
hematemesis	Yes	No	No	No

As nouns, hematemesis is a hyponym of puking; that is, hematemesis is a word with a more specific, narrower meaning than puking:

puking: the reflex act of ejecting the contents of the stomach through the mouth
hematemesis: vomiting blood

Other hyponyms of puking include rumination, haematemesis, hyperemesis.

puking (noun)	hematemesis (noun)
the reflex act of ejecting the contents of the stomach through the mouth	vomiting blood