It’s not that they’re truly synonymous but that each also implies the other. If it’s true that he’s dead, then it’s also true that he died and vice versa. So it seems like they mean the same thing because if ypu say one, it can be taken for granted that the other is necessarily also true.

But even that’s not 100% - it’s possible that “he died” is true but “he’s dead” is not, since he might’ve been revived. That illustrates the fact that they actually each communicate something different - “he died” is an experience through which he went at some point, while “he’s dead” is the state he’s in right now.

So again, they broadly communicate the same thing since saying one implies the other as well, but they don’t actually mean the same thing.