insignia vs badge

insignia and badge both are nouns.

Word	Noun	Adjective	Verb	Adverb
insignia	Yes	No	No	No
badge	Yes	No	Yes	No

As nouns, badge is a hypernym of insignia; that is, badge is a word with a broader meaning than insignia:

insignia: a badge worn to show official position
badge: an emblem (a small piece of plastic or cloth or metal) that signifies your status (rank or membership or affiliation etc.)

insignia (noun)	badge (noun)
a badge worn to show official position	any feature that is regarded as a sign of status (a particular power or quality or rank)
	an emblem (a small piece of plastic or cloth or metal) that signifies your status (rank or membership or affiliation etc.)

insignia (verb)	badge (verb)
	put a badge on