Mastodon is just a bunch of isolated copies of the same app talking to each other. There is no notion of a shared identity, each server's admin is effectively a king over your account, etc. It's a fragmented patchwork of isolated sites that forward messages.
With AT, there is just one global network. Like the web. You don't post "to" someone's isolated copy of an app. You post to your own folder, and every interested app can aggregate your post.
It's a bit like email vs RSS. Very different shapes.
To give you a concrete example to ground it. Blacksky is a fork of Bluesky. They're setting up their own server with different moderation policies and are unbanning some people that Bluesky has banned (https://bsky.app/profile/rude1.blacksky.team/post/3mcozwdhjo...). However, Blacksky posts still exist in the same "world" as Bluesky--they are not an isolated fragment.
Thanks to AT, Blacksky and Bluesky are just two different prisms through which you see a single network. Whereas with Mastodon, every app is its own network, with some limited message passing between them.