self-importance vs conceit

self-importance and conceit both are nouns.

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

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

Other hypernyms of self-importance include conceitedness, vanity.

self-importance (noun)	conceit (noun)
an exaggerated opinion of your own importance	the trait of being unduly vain and conceited; false pride
an inflated feeling of pride in your superiority to others	a witty or ingenious turn of phrase
	an artistic device or effect
	an elaborate poetic image or a far-fetched comparison of very dissimilar things
	feelings of excessive pride