self-importance vs vanity

self-importance and vanity both are nouns.

Word	Noun	Adjective	Verb	Adverb
self-importance	Yes	No	No	No
vanity	Yes	No	No	No

As nouns, vanity is a hypernym of self-importance; that is, vanity is a word with a broader meaning than self-importance:

Other hypernyms of self-importance include conceit, conceitedness.

self-importance (noun)	vanity (noun)
an exaggerated opinion of your own importance	low table with mirror or mirrors where one sits while dressing or applying makeup
an inflated feeling of pride in your superiority to others	the trait of being unduly vain and conceited; false pride
	the quality of being valueless or futile
	feelings of excessive pride