What is the difference between updating an 'A' record and a 'CNAME' record in Cloudflare?

An 'A' record directly maps a hostname to an IP address. A 'CNAME' record (Canonical Name record) maps a hostname to another hostname. If you're using a 'CNAME' record, you'll need to update the target hostname to point to the server with the new IP address, rather than the IP itself.