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.