I am working on an updated ABOUT page for Graz'zt for when I swap blogs, and as part of that, I'm including a small section to (hopefully) easily explain the differences between demons and devils within the cosmology. Because I am obnoxious and can't quiet the little voice that goes "um, actually" when I see demons and devils being referred to interchangeably.
For player characters, alignment is descriptive rather than prescriptive. This is not the case for demons and devils (...sort of), where the difference between Lawful and Chaotic is what fundamentally separates them from one another.
A chaotic society is one in which the strong rule—might determines right. In this societal structure, it is nearly impossible for an incompetent person to become leader because strength is the only virtue that matters, and if you are strong enough to rule, you are capable of killing those in your way. Iggwilv’s excerpt from the Demonomicon speaks explicitly to this.
“As creatures of chaos incarnate, demons have no unified culture. The only modicum of order that exists for a demon is the one imposed upon it by a more powerful demon—and the moment that a demon rises high enough to assert its authority on others is the very moment that rivals begin to subvert that authority.” — The Demonomicon of Iggwilv
Graz’zt is powerful enough to maintain three layers of the Abyss in his kingdom, and no one has yet been able to best him in cunning or strength. Therefore, he leads.
On the other hand, in a lawful society, the laws determine order; rules come first, the actual competence of the leader second. This is where we see a concern for traditions, proper forms being obeyed, and the idea of some kind of established legitimacy.
Not everyone may like it, but Asmodeus rules in Baator because the structure, rules, and traditions say so. The Lord of Lies is King of the Hells because people believe it; they follow the rules, even if they don't like it. His rivals may scheme and plot, but they cannot replace him just by defeating him in combat. There are structures to be observed, and if the other powerful movers and shakers say that you are not in charge... you are not in charge.