If we specify the golden age of Athenian democracy, which lasted for most of the fifth century BCE, and the Roman Republic, which lasted from the expulsion of King Tarquin in 509 BCE to the establishment of the empire by Augustus in 27 BCE, the first point to make is that the Roman Republic lasted for five times as long as the Athenian democracy and changed significantly during that time. In 509 BCE, Rome was essentially a city-state like Athens. By the end of the Republican period it was a vast, sprawling empire without an emperor. The differences between the two civilizations therefore increased as the Roman Republic grew. In Athens, all free adult men had voting rights. In Rome, active participation in politics was limited to a fairly small number of Patrician and wealthy Plebeian families, and political life became notably more corrupt towards the end of the Republic.

Both cultures gave free adult men a high degree of personal liberty. However, Roman society made it easier for a slave to attain wealth and influence, and much easier to become a free man, a rare event in Athens. Women also enjoyed a much greater degree of freedom in Republican Rome and the political influence of wealthy women grew over the course of the Republic. Both Athens and Rome produced remarkable corpuses of great literature, though the great classics of Roman literature by such poets as Virgil, Horace, Catullus, Propertius, Tibullus and Ovid, along with the prose of Cicero and Julius Caesar, were overwhelmingly produced in the last years of the Republic or the first years of the Empire.

By Greeks, I am assuming you are talking about Athens in the fifth century BCE. They shared a number of similarities with the democratic practices of the Romans, particularly during the period of the Roman Republic. However, there were also many significant differences.

Similarities

The right to vote in both ancient Athens and Rome was limited to adult male citizens. Women and slaves had absolutely no say in political matters. Overall, though, both emphasized the role of election and the fundamental role of the people in setting the terms of a government's legitimacy. For example, in the Roman Republic, laws were passed by being approved by a popular assembly. The Senate could make decrees and manage the affairs of government, but it was up to the voting citizens to actually pass a law. Likewise, in ancient Athens, laws had to be passed by a popular vote.

Differences

Democracy in fifth century Athens was much more direct than it was in Rome. Athens, having a much smaller population, was able to practice direct democracy. Citizens voted on issues as popular referenda. Rome, on the other hand, had a huge population. Voting directly on the functions of government would have been too cumbersome and time-consuming. Therefore, the Romans elected representatives to manage these issues.

In Greek democracy, the poor had a lot more authority than did the Roman poor. Citizens of all classes served on Athenian juries, including the poor. Ordinary Greek citizens could find themselves in influential bureaucratic and administrative positions, as these jobs were assigned by lottery. In Rome, these administrative positions were made by appointment or through elections.

The similarities of the two governments are:

1. Both people groups did not espouse any type of king. The Greeks had a famous king who was greatly loved, Codrus. When he died they realized that no one would be better. So, they did away with kings. The Romans did away with kings, because they could not stand the evil of Tarquinius. So, they both did not have kings, but for different reasons.

2. They still had magistrates that rule on behalf of the people. The Greeks had archons and the Romans had consuls.

3. Both groups had a sophisticated laws.

Here is one difference to get you started.

1. The Greeks had something called ostracism, and the Roman did not have this practice.

The major similiarites are:

1. Both systems allowed for some amount of input from all or most of the male citizens of the society.
2. Neither was a monarchy, which is what just about every other society was in those days.

So both of these types of government are pretty unique for their time and that's a very important similiarity.

The major difference is that the Greek democracy gave more power to more people than the Romans did.  The Romans had a direct democracy that got to vote on a lot of laws and have a part in carrying out the laws.  Roman laws were made much more by elected officials (Senate) and carried out by government officials like consuls and censors rather than by more ordinary people.

So both were very democratic for their times, but the Greek system was more of a direct democracy and the Roman system was more like our representative democracy.

By Greek Democracy, I assume you mean Athenian democracy. That government was a true democracy in which all adult free males were citizens and entitled to participate. Although early on, the aristocrats controlled the government, Solon arranged for all classes to be represented in the government. Attendance at the assembly was mandatory for all males who were able to attend, similar to jury duty today. Slaves were sent into the agora with ropes dipped in red dye. They marked the robes of any free adult male who was not at assembly, which served to publicly humiliate him, and also subjected him to a fine.

In Rome, the old Patrician families had control of most government functions. They sat as the Roman Senate. The lower classes, or plebeians, were allowed to elect a Tribune who had the power of veto over any decisions of the Senate which he considered unfair. There were a number of other Government officials, such as the Pontifex Maximus, who set the official calendar and decided which days were appropriate for business; a Censor who prosecuted those Senators guilty of corruption; and the Concilium Plebis, comprised only of Plebeians, which eventually had full control over domestic affairs.