How did Rome's republic differ from Greece's democracy? How were they similar?
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.