There really was no singular "Greek" model of government. Greece evolved as a series of independent poleis, or city states, which each had their own models of governance, ranging from democracy, through dictatorships, to kingdoms, to oligarchies. Most of Greece had been conquered by Philip and Alexander of Macedon, and then the cities became quasi-independently governed parts of Macedonian successor states, and were eventually conquered by Rome during the time of the Republic, although many retained councils responsible for local administration.
Imperial Rome was an empire ruled by an Emperor, with uniform laws, monolithic military and administration, and a vast territory.
In the Greek model of government, the city-state or polis remained independent. Due intitially to geography, these city-states grew and thrived as little individual kingdoms. the two most important of these were Sparta, a military oligarchy, and Athens where democracy developed. They only united once in history to fight against Persia as they held their independence and freedom as their highest value. Though significantly smaller in number, they defeated this mighty empire. The Roman Empire, however, became a huge military machine that was impregnable. They conquered and demanded allegiance from all their subjects. Once under Roman control, they paid taxes to Rome and either submitted to Roman rule or faced the consequences. Greece, having been weakened by civil war, became Rome's territory. Neither Democracy nor the Republic form of government that was developed in Rome would be feasible in such a massive empire as was Rome. It eventually proved to large to be properly governed by an emperor which was one of the contributing factors in its demise.