It largely depends on what city state we are talking about. People at this time didn't identify as Greeks they were Thebans, Ionians, Athenians, Thracians and so on.
Syracuse and Corinth had well developed democratic systems and many of the other city states had democratic components. Even the militaristic duel monarchy of the Spartans had some democracy. The Spartan Ephors were elected and had enough power to exile kings if desired. Let's look at the most commonly cited city state though, Athens.
Athenians utilized a direct democracy compared to Roman republic and everything would be voted upon. Who would lead the armies, what kind of trade arrangements, who the diplomatic envoys would be; pretty much everything. Romans elected specific individuals to handle regional business like a representative of the people, the senators and provincial governors.
Athens was slightly more equal then the Roman system. Under the Athenian law all free citizens technically had equal rights in the government. In Rome the Patricians, Equestrians and Plebeians had strictly defined roles both legally and legislatively.
The reverse of the above point were the inclusiveness of the two systems. Athens had draconian regulations on who qualified as a free member of the city state. Rome comparatively welcomed a large swath of people and actively sought to latinize the frontiers.
Rome had two consistent political parties. Politics would be marked by conflict between the Conservatives and the Populares. Athens comparatively had many political blocks that were constantly evolving and changing beliefs.
The most distinct difference between the system is probably the adaptability of them though. The Roman system was extremely complex, but was constantly changing and adapting to meet the changing times. The Athenian system would become bogged down as time progressed and would essentially become little more then another oligarchy towards its end.
