One difference: With an 0-day you know your target can't have done anything to specifically mitigate that vulnerability. If a vulnerability is well known but still unpatched by the vendor, a potential target can take their own steps to protect themselves.
For example, if you're running some ancient mailing list software that you know has an unpatched XSS vulnerability, you can have your front end servers scan for attempts to exploit that and abort the requests. Or lock it down with a CSP policy. Or if you know your image manipulation library has tons of vulnerabilities, you could run it in a locked-down sandboxed environment where exploitation doesn't get the attacker much of anything.
For example, if you're running some ancient mailing list software that you know has an unpatched XSS vulnerability, you can have your front end servers scan for attempts to exploit that and abort the requests. Or lock it down with a CSP policy. Or if you know your image manipulation library has tons of vulnerabilities, you could run it in a locked-down sandboxed environment where exploitation doesn't get the attacker much of anything.