What is the difference?
Sign In with Apple allows one to sign in to Apps and websites with your current Apple ID. Using Sign In with Apple precludes having to sign in with an existent social media account or creating a new password.
Hide my Email is part of Apple’s iCloud+. Hide My Email allows one to create unique random email addresses that forward to your personal email. Hide My Email can be created for any App or website that requires an email address for registration or communication. Hide My Email can be used in Apps and Websites that do not support Sign In With Apple.
Both Sign in With Apple and Hide My Email are good ways to protect your anonymity online by protecting your personal email address.