illegitimate child vs offspring

illegitimate child and offspring both are nouns.

Word	Noun	Adjective	Verb	Adverb
illegitimate child	Yes	No	No	No
offspring	Yes	No	No	No

As nouns, offspring is a hypernym of illegitimate child; that is, offspring is a word with a broader meaning than illegitimate child:

Other hypernyms of illegitimate child include issue, progeny.

illegitimate child (noun)	offspring (noun)
the illegitimate offspring of unmarried parents	any immature animal
	something that comes into existence as a result
	the immediate descendants of a person