They both have the same meaning . However , us English speakers don’t really use gainsay. Naysay is a better choice.

According to my dictionary: Gainsay (verb) to deny, contradict dispute (noun) denial Naysay (noun) a refusal Nay-sayer - someone who refuses to do or agree to something. You sometimes hear nay-sayer, but these aren't common words. I can't remember using them.