brilliance vs elegance

brilliance and elegance both are nouns.

Word	Noun	Adjective	Verb	Adverb
brilliance	Yes	No	No	No
elegance	Yes	No	No	No

As nouns, elegance is a hypernym of brilliance; that is, elegance is a word with a broader meaning than brilliance:

brilliance (noun)	elegance (noun)
a light within the field of vision that is brighter than the brightness to which the eyes are adapted	a refined quality of gracefulness and good taste
the quality of being magnificent or splendid or grand	a quality of neatness and ingenious simplicity in the solution of a problem (especially in science or mathematics)
unusual mental ability