What is the most distinctive political difference between the Greeks and Persians?
The greatest difference between the Persians and the Greeks was undoubtedly in form of government. Persia was governed by a king, who was a quasi-divine figure. Moreover, he unified Persia, a huge and diverse empire. The Greeks were very different.
The Greeks did not have a divine king that governed all of them. In fact, by the time of Aristotle, the Greeks had 1,100 independent cities. This means that each city ruled themselves. This is a radical difference. Moreover, these cities had various forms of government - tyrannies, democracies, oligarchies, and even mixed constitutions like Sparta. For example, Sparta had two kings, five ephors, and a council, called the gerouisa. All of this is in stark contrast to Persian.
Finally, I should mention that during the classical era, the Greeks did not really like the idea of a king.