What were some similarities and differences between the democracies of ancient Rome and ancient Greece? Which is closer to the American form of...

The word
democracy (demos, 'people' and 'kratia' power or rule)
was coined about 2500 years ago to establish a classless government -- one that would not be
dominated by the aristocracy, and would include what we would understand to be the middle class
(slaves, of course, had no influence in government.)  The new Athenian democracy was composed of
three institutions, the ekklesia, or Assembly, which was the sovereign
governing body of Athens, similar to our concept of the Senate.  The second was the
boule, which dealt with day to day matters, and might be considered
analogous to our House of Representatives. It was here that the concept of a direct
democracy evolved. The third was the dikasteria, or the courts.
 This system of governance lasted about 200 years, at which time it reverted back to an
aristocracy.

At the same time in Rome, the concept of a Republic
(res, 'concern' and publica, 'people') evolved.  The
aristocracy at the time was attempting to take absolute political power from the Etruscan Kings.
However, they needed the assistance of the rest of the population, promising them some political
say in the only governing body, the Roman Senate.  This was done, but Rome established a
representative democracy, where some of the people could vote for a
representative in the Senate (a Tribune.) The Roman Republic lasted about 500 years, at which
time the Republic devolved into the dictatorship of the Emperor.

Since, in
theory, the United States splits its government into seperate branches,  the legislative branch
having an upper and lower house, and the judicial branch dealing with court matters, it is
similar to Athenian Democracy.  However, its elections are for candidates that represent the
people, so it functions similar to the Roman Republic in that respect.  In either case, these
ancient experiments in "people's governance" failed.