regardless vs no matter

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

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

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

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

Other synonyms of regardless include disregarding, disregardless, irrespective.

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

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