I don't think Zeus ever argued with Dionysus. (Do tell if you know otherwise). I think this is because Dionysus is really as powerfull as Zeus, but not interested in managing the world. So Zeus is careful not to offend him - and so is able to stay on as chief god.
Dionysus is nowhere near as powerful as Zeus. For one, he was born half-mortal, and elevated to the pantheon later, as with Heracles and other common half-mortals.
It was an unarguable fact that there was no god or goddess stronger than Zeus. Through his strength and his cunning, he had ensured that not only were the Titans he overthrew permanently laid low, but that no child born to him by a Titan would overthrow him, permanently securing his seat of power. Zeus had proclaimed that if all the Olympians together tied a rope to the highest peak of Olympus and hung Zeus from it, if all of them held onto the rope and tugged they couldn't drag him down. Zeus' thunderbolts were completely indestructible and could shatter any material in existence, except his aegis which he gave to his favourite child, Athena.
Certainly Zeus could be deceived, so he wasn't entirely omnipotent or omniscient like the Abrahamic God. But his authority could never be challenged because he had cemented his status by securing his past and his future from being usurped by any other opposition. He was the son of a Titan and had defeated them, and none of his sons, not Apollo, Ares, Hermes, or Dionysus, could ever achieve his power to overthrow him. The Olympians all knew they were subservient to Zeus, and so they did their best to keep him placated.
The only deities, if you could call them such, who had any power over Zeus were the Fates, to whom Hesiod admits that not even Zeus can control the course of life and death. Zeus gives to them the greatest honour, because he cannot reverse history, or destiny, any more than humans. But, being more personifications than gods, the Fates are not the rulers of the world, and in the war against the Giants they were defeated in battle, but Zeus was not.