Nameservers and DNS records are related, but they are not the same thing. Confusing them is one of the easiest ways to break a website or email during a move.
Nameservers Say Who Is in Charge
Nameservers tell the internet which DNS provider is authoritative for the domain. They are set at the registrar.
DNS Records Say Where Things Go
DNS records live inside the DNS zone. They point the website, email, subdomains, and verification checks to the right places.
Changing Records Is Smaller
If you only need to move the website, changing the A record may be enough. Email and other records can stay untouched.
Changing Nameservers Is Bigger
Changing nameservers moves control of the whole DNS zone. You must recreate important records at the new provider before switching.
When in doubt, move the smallest thing necessary. It reduces the chance of accidentally breaking email or third-party services.