The difference between the Greek democracy and the Roman republic is that in Greek democracy, each man got a vote; in the Roman republic, each man voted for a representative.
The Greek democracy worked in the polis or city-states. There, all the Greek citizens were entitled to vote and participate in the political matters, and it was a duty. The city-states considered slaves to be necessary so the citizens could focus on their obligations. On the other hand, in the Roman republic, citizens could elect a representative such as a senator and the senators were the ones to elect a ruler, similarly to the U.S. election system.