no matter vs regardless

no matter is not an adjective while regardless is an adjective.

Word	Noun	Adjective	Verb	Adverb
no matter	No	No	No	Yes
regardless	No	Yes	No	Yes

As adverbs, no matter and regardless are synonyms defined as:

no matter and regardless: in spite of everything; without regard to drawbacks

Other synonyms of no matter include disregarding, disregardless, irrespective.

no matter (adjective)	regardless (adjective)
	(usually followed by `of') without due thought or consideration

no matter (adverb)	regardless (adverb)
in spite of everything; without regard to drawbacks	in spite of everything; without regard to drawbacks