Wow, this thread is one of the reasons I wanted to join this forum.
I believe every is well said, but I would like to provide a perspective. I believe the Emperor, the Baron, and the Duke, even Paul are examples of leaders and how they lead. The Emperor foundational power was on the fact he possessed the Sardaukar. He operated with military power and pretty much military power alone, although it was a force to be reckoned with. He had the BS and the Guild whispering in his ear and he was waffled by these forces. He knew he could not maintain power with out the Guild, BS, or consent of the Landsraad. The Emperor is our military leader and suffers the weakness of military leaders: the support of the people around them and the singularity of their focus, War. He was also the seat of Monarchical power.
The Baron is our mad despotic ruler, like Ghanis Khan. He lives to consume others and his primary power is in his brutality. He acts in this way and uses it with Machiavellian wit. He is thrust into the plan do to his hate of the Atreides and the hope of gaining power. He is the simple and convenient tool.
The Duke is our charismatic leader. He wields the power of his people and Landsraad loves his fair dealing and strong devotion to loyalty. With this "fairness" he attract forces which are forged by our other leaders. The Duke then begins to develop a military power, albeit small at this time. This is why I think he was a threat. He was at first only charismatic. Now he is becoming a military force. The Emperor saw the Duke as a threat, because if he had a military force equal to him, then he was more powerful because he could wield it with the consent of the Landsraad. I think this is proved by the fact that the Emperor moved him to Dune, to remove his charismatic influence away from his power base. Operating the old axiom, Never fight a shark in the water. (hehe I know that was a hare punny, sorry).
Along the subject of leaders, this brings me to Paul. In terms of leadership, he is all leaders in one. Military power base, Economic leader (the spice must flow), Fanatical religious (Messianic), Monarchical marriage passing of power (Irulan), and brutality stored in his Jihad.
I know this seems over simplified, leadership styles as a reason. Yet, I feel it's the one singular facet that explains why Atreides scared the s#!+ out of Emperor.
"The fool who persists in his folly will become wise." William Blake
"Duncan, have I not told you that when you think you know something, that is a most perfect barrier against learning?"