They are very close in meaning. Tony is gone=Tony is not here but he was. This is about location. Tony has gone= Tony left earlier. This is about the action of leaving. Bùi is incorrect. You absolutely can say "Tony is gone" and it doesn't have to mean he is dead (but it could).

如何区分托尼走了和托尼走了？

In English, the verb "go" usually requires a destination/purpose. Therefore, it'd be much more natural to say "Tony has gone to work" or "Tony has gone out for lunch". "Tony is gone" is an old-fashioned way to say he has passed away.