How did greek and roman versions of democracy differ?

The Rome was not a democracy, but a republic. A republic (res publica or thing of the people) is a form of government in which those with a vote elect representatives to act for them in a legislative body.

The Greeks had democracies in Athens and some of their other city-states. A democracy (from Greek Words meaning rule by the people) is a form of government in which those with a vote meet directly as a legislative body to make decisions and pass laws.