what are the differences between the political structures of government between the Greek polis and the Roman empire?
This is a good question. First, you need to keep in mind that there were many Greek cities and each city had its own political structure. So, during the fourth century B.C., Athens was a democracy where the people had the power to vote and shape their future. Sparta during that same time had a very different political system. For instance, they had two kings, a groups of elders, and ephors who watched over everyone. Other cities were oligarchies, where the local aristocracy ruled, and still other cities had tyrants. So, diversity is the point.
When it comes to the Roman Empire, we can say that it was very different from the Greek cities for a few reasons. First of all, the Roman Empire had an emperor who ruled everything and everyone. From the point of view of the law, his legal opinions were de facto laws. He had no accountability and he could pretty much do what he wanted. Even the senate was only advisory and as time progressed became less influential.
Second, for many periods of the Empire, succession was not based on voting, but on lineage. For instance, the Julio-Claudian line ruled in Rome for a long period of time. The same was true of the Flavians.
Another difference in political structure was that the Romans carved out various political offices based on functions. So, the quaestors took care of finance. The praetors took care of the law. The aediles looked over the maintenance of the city. The censors conducted the census. These position did not exist in the Greek world. Rome needed these things, because they ruled an empire.