vegan vs vegetarian

vegan and vegetarian both are nouns.

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

As nouns, vegetarian is a hypernym of vegan; that is, vegetarian is a word with a broader meaning than vegan:

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

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