Password gets sent to the server. Private key does not get sent to the server, and authentication protocol can be forwarded safely, enabling things like ssh-agent, FIDO SSH keys. This is solvable with passwords (see: augmented PAKE), but `PasswordAuthentication` isn’t that. But there are also settings where none of this really matters, yes, to be fair if that’s what you’re referring to.
SSH certificates are cool too.
password doesn’t get sent to the server, hash does.
but I guess that the hash is always the same unlike with pub priv key mechanism.