If I were talking about Passkeys and comparing them to client certificates, even though I don’t know much about client certificates in practice, I would say:

• Passkeys can be installed in your password manager, which handles securely syncing it to all of your devices
• Websites can make it very easy to create or log in with a passkey
• Far more websites support passkeys
• Websites can support multiple passkeys per user
• The user experience is far better with passkeys
• Even if your password manager isn’t installed on a given machine, you can still log in with a passkey via your phone, so long as both devices have bluetooth enabled. This allows you to log in on an untrusted device, like a library computer, without exposing your password (though unfortunately that would still result in that computer having access to the session and being able to modify account settings - best practice would be to log out when you’re done and then, from a trusted device, confirm that you were logged out / log out of all devices.)