You can only post through your account on Twitter if you are logged into your account. That is too say: The twitter API call authenticated you via some Security Service. That is how they know it’s your note. Cause it was posted via your credentials.

However, in nostr your credentials are embedded into the note itself via the “sign” field in NIP-01. This allows someone, anyone, to verify my note simple by looking at my note. They do not need to trust the authentication of any account.

If I post my signed nostr note on Twitter, then twitter will “act” as a relay in this instance. Twitter will own a copy of my note. But anyone can verify that note was signed by me without depending on Twitter API at all. In the real world though, using Twitter as a relay is useless.