It is interesting that you mention ancient Greece, Persia, and the United States. These three countries are related in very interesting ways. Since you ask for differences, let me name a few of them.
First, Ancient Greece for the most part was democratic. Not all cities were democratic, but enough of them were democracies that we can say that democracy was a distinguishing mark of Greece, or at least a lasting contribution of Greece. Persia, on the other hand, was a divine kingship. Also we can say that the Greek cities did not have divine kings (apart from mythology). From this point of view, the Greeks and Persians were very different. The United States inherited democracy from the Greeks and brought it to a modern form.
Second, the Persians were imperialistic. They sought to conquer people, and they were good at it. To put it another way, the Persian king was the king of an empire. The Greeks never formed an empire. Ancient historians say that there were a thousand cities. Some formed alliances, but many of them were independent. This was very different than Persia. Some would say that America is imperialistic, not in the form of attacking and conquering, but conquering in a different way, through fear, economic advantages, popular culture, and the like.
Thinking about these points should get you started.