vegetarian vs vegan

vegetarian and vegan both are nouns.

Word	Noun	Adjective	Verb	Adverb
vegetarian	Yes	No	No	No
vegan	Yes	No	No	No

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

vegetarian: eater of fruits and grains and nuts; someone who eats no meat or fish or (often) any animal products
vegan: a strict vegetarian; someone who eats no animal or dairy products; may also avoid using animal products (e.g. leather)

vegetarian (noun)	vegan (noun)
eater of fruits and grains and nuts; someone who eats no meat or fish or (often) any animal products	a strict vegetarian; someone who eats no animal or dairy products; may also avoid using animal products (e.g. leather)