To me, “he died” puts an emphasis on what the person actually went through. To die is to experience the process of dying. “He is dead” puts the emphasis on his current state, not on the transition from life to that state. Linguistically, I consider dying to be the process and death to be the result. You die once, but you stay dead forever (medical resuscitation notwithstanding).

I have no clue how many other people think of the phrases like that, but that’s the rhetorical distinction I draw between the two.