out of wedlock vs illegitimately

out of wedlock and illegitimately both are adverbs.

Word	Noun	Adjective	Verb	Adverb
out of wedlock	No	No	No	Yes
illegitimately	No	No	No	Yes

As adverbs, out of wedlock and illegitimately are synonyms defined as:

out of wedlock and illegitimately: of biological parents not married to each other

out of wedlock (adverb)	illegitimately (adverb)
of biological parents not married to each other	of biological parents not married to each other
of unwed parents	in a manner disapproved or not allowed by custom