Sign in with Apple is Apple's SSO, like Sign in with Google. Services have to support this one explicitly, and it already had a special subdomain, so the specific subdomain is simply changing.

Hide My Email is the manually generated ones, for websites that accept an arbitrary email address. This is the one where it's valuable for the relays to be identical to genuine iCloud addresses, otherwise websites could try to block it and force you to use a more revealing email address, undermining privacy.