What were some strengths and weakneses in the political system of the Greek polis and the Roman state?
This is a huge question, because both the Greeks and Romans lasted a long time and were complex people. Let me deal with the Greeks first and then the Romans.
As for the Greeks, I will use the city of Athens as an example. The strengths of Athens were chiefly due to the fact that they made all people active citizens. In other words, they had a radical democracy. People got involved and took ownership. This created people who intensely cared about their city. This led to great advances in architecture, art, and philosophy. The weakness was that at times there was no strong leader. This could lead to a deadlock. Think of America now. Little gets done, because power is too dispersed.
As for the Romans, I will speak of the Republic. The Roman Republic has a mixed constitution. They possessed people with great power, like the consuls, but there were always two of them. In fact, they always had a plurality of people in power together. This was a benefit, because it created natural accountability. No one person could take over. However, there were weaknesses as well.
The main weakness was the Romans were too competitive. They lived to fight, in a sense. So, when they defeated their enemies, they fought each other and this created many bloody civil wars. The Romans killed themselves. More to the point, their government was structured to be competitive.