one of the, if notthe, strongest military in all of ancient Europe.

This absolute devotion to traditional principles also had its downside. Sparta engaged in a yearly war against one of its neighboring city-states in the Peloponnesus—Messenia. Spartans would capture adult males from Messenia and force them to work in the fields and domestic economy. These “helots,” or Spartan slaves, undergirded the entire Spartan economy. Because Spartan men devoted all of their time to martial pursuits, they had no time to manage their own households or engage in other, more urbane activities. Thus, Spartan citizens had to maintain their strength and superiority in combat, as the helot population outnumbered the native Spartan by a factor of 10 to 1. If the helots were to successfully revolt, not only would it deprive the Spartan polis of its source of labor power, but it would also threaten the entire social structure on which the Spartan martial mentality had been constructed. This dependency on slave labor restricted the Spartans from expanding outward, as all of their attention had to be devoted to maintaining the slave population at home.

In Rome, Republican values also stressed the importance of tradition and especially filial piety. The Roman Republic early on adopted the political philosophy of patria potestas (“fatherly power”), which stressed the single authority of the male head of household. Fathers had so much authority in governing their family that they were legally permitted to execute children who proved to be especially insubordinate.

This level of patriarchy had some benefits to the development of the early Republic. The government of Rome was essentially modeled after the authority of the father in his own household. “Elders” were senators and “father figures” were patricians, and the Roman state could flourish only if these father figures lent their active and continuing support. Most of all, this philosophy clearly distinguished the lines of authority in the state, allowing for early stability and the streamlining of decision-making in law.

Ultimately, however, the corruption that was bound to proliferate in such a striated form of society led to intense social dissatisfaction with the distribution of power. The poorest classes of Republican Rome were finding new champions among progressive aristocrats, who sought to reform the government according to more liberal principles. Most famously among these aristocrats were the so-called “Gracchi”—Tiberius and Gaius Gracchus—who attempted to reform property law so as to reduce Roman reliance on slave labor and to limit the power of the wealthiest patrician families. The Senate, primarily staffed with wealthy patricians who stood to lose much if the Gracchi reforms went through, ordered Gaius and his supporters to be ruthlessly purged. Therefore, just as the principles of patriarchal order provided stability to a growing empire, it also sowed the seeds of intense corruption and political intrigue.

Let's start with the Greek polis. As each city-state operated under a different political system, it is necessary to speak generally to answer your question. An advantage that they had was that, with relatively small and localized populations, each polis could adapt its system to serve the particular needs and customs of its people. For instance, Spartans highly valued military accomplishment and the wisdom of elders. Therefore, they had a government made up of warrior kings and a council of elders. Meanwhile, Athenians valued citizen involvement and therefore established a popular democracy.

A clear disadvantage of the Greek poleis was their small size. By being surrounded by other Greek states and neighboring empires (i.e. Persia), the poleis often found themselves at war. Furthermore, poleis often experienced revolution, rebellion, and overthrows of power. Frequently, a faction would arise in a polis that would overthrow the current political system and replace it with their own, often with a tyrant at its head.

Now let's turn to Rome. During the Roman Republic, power was effectively spread among members of the Senate, the two Consuls, and the tribunes. This division of powers and responsibilities served to make sure that no faction or person was able to seize too much power. It also ensured that different populations within the Republic had representatives in government.

A disadvantage of this system had to do with its complexity. With so many different elements involved in passing and implementing laws, the government could be slow to effectively respond to threats and urgent issues. It was also common for ambitious leaders to compete with each other in their quest for power in ways that damaged the integrity of the Republic overall.

During its time as an empire, Rome was led by a series of powerful emperors. An advantage of this system was that decisions were made and implemented quickly as they relied only on the wishes and orders of one man rather than a complex political apparatus. The disadvantage of this was that there were few safety measures. While Rome had many intelligent and enlightened emperors, it also had its share of cruel and mentally unstable emperors.

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.