The typical distinction drawn between ancient Greek democracy and modern democracy is the degree of participation involved in Greek democracy. In Athens, citizens took part in government through the ekklesia, a popular assembly responsible for declarations of war, taxation and the election of some political/judicial offices. The ekklesia met frequently throughout the year and all male citizens were welcome to participate. Furthermore, even the poorest citizens were encouraged to participate. Citizens who took part in the ekklesia for a full day were remunerated, so that poorer citizens would not have to give up a day's work to take part. Voting was done by a show of hands. Male citizens could also be allotted a seat on the Council of 500 (the Boule). Its principal responsibilities were determining the agenda for the ekklesia and most likely the day-to-day governing of the city. Any male citizen could have a seat on the council. Its members were chosen by lottery and served a term of one year. A citizen could serve up to two terms on the Boule, ensuring a regularly rotating membership and greater access for the citizen body.

The political lottery was particularly important in classical Athens. Most political offices in classical Athens were chosen not by election or appointment, but by lottery. Many of these offices resembled public works jobs that are sponsored by governments today, such as student jobs in museums, parks, etc. For example, one office was maintaining and repairing windows around the city, so that they would not fall onto the street and injure any bystander. There were exceptions for certain political offices, such as military generals (the strategoi), who were elected by the ekklesia. However, according to most democrats, the lottery was the most free and fair way to decide political offices. Otherwise, the social elite would dominate them, since they could employ the most resources in order, for example, to win an election.

Although the Athenians did not have many elections, they did have a peculiar form of "anti-elections" called ostrakismos, in which the person with the most votes (with at least a minimum of 6000 votes) was exiled from Athens for a period of ten years. However, after he served his exile, he could return to Athens with his property and social status intact. Many other democracies, such as the Second Syracusan Democracy, practiced their own forms of ostrakismos. It was a safe way to ensure that no citizen became too politically powerful to threaten to overturn the democracy. However, it gradually petered out of use in Athens, probably because it was fairly ineffectual.

It is also important to note that the majority of the population in ancient Greek democracies was excluded from participating, namely women, slaves and foreigners. As far as I know, women did have citizenship in Athens (at least after Perikles' time), but could not participate in political affairs.

So, to sum up, the main difference between modern democracies and ancient Greek democracies is that the ancient Greeks did not delegate political decision-making to representatives. Rather, it was undertaken by the citizen body through active participation in the ekklesia and boule. Additionally, the Greeks stressed the importance of the lottery in doling out political offices. The practice of ostracism is also peculiar; however, its importance should not be exaggerated, since it was rarely used.

As for the Roman Republic, I'm a little less certain. The Romans did not have much of a democratic system. I know that the Greek historian Polybius claimed that the Romans had a "mixed" constitution that combined the best elements of democracy, aristocracy and monarchy. Citizens' votes were weighted according to their property value, such that the vote of the wealthiest section of Roman society, despite being numerically smaller, far outweighed the poorest. The Roman elite therefore dominated the senate, consulship and tribunate.

Sources: Aristotle, Athenaion Politeia, is probably the most comprehensive primary source on the Athenian political system. Polybius, The Histories, book six especially, talks about the Roman constitution.