Ultimately, if you're going to compare Ancient Greece against the United States, the most fundamental and defining difference lies in the reality that Ancient Greece was not a single unified polity (the way the United States is), but rather was made up of various city states, which would have had different governments and government types.
Things become more complicated still when you factor in the larger trends of history and the way that government changed and evolved over time: for example, the Ancient Greek democracies tended to evolve out of a more gradual transition across time. The Ancient Greek city states that emerged out of the Greek Dark Age were usually aristocratic in form, and many of these aristocracies fell under the sway of political strongmen, or tyrants (this is where the word tyranny came from). Ancient Greek democracy evolved out of this context, after the downfall of these tyrants.
In any case, with a question such as this one, the most natural point of comparison lies in democracy itself (most famously associated with Ancient Athens). Ultimately, modern democracy is actually very different from the Athenian vision of democracy. Whereas modern democracy is representative in nature, with citizens electing politicians to govern for them, Athenian democracy was a direct democracy, with key policies and decisions voted on directly by the citizen body itself. Furthermore, influenced by the history of the tyrannies, Athenian democracy contained various safeguards aimed specifically at preventing any one citizen from gaining too much political power (and becoming a threat to the democracy). For example, while we choose political leaders by election, the Athenians were actually known to use randomized lots to select key political leaders.
That being said, in other respects, Athenian democracy was quite limited compared to government today. Ultimately, what you must remember is that, while citizens in the Ancient Greek context exercised remarkable agency and political power, the Ancient Greeks also had a very narrow concept of citizenship, so that only a very small subset of the larger population had political power. This contrasts dramatically with modern democracy, which is far more inclusive where the entire population is concerned.
That being said, there are similarities between the two governments as well. In many respects, both Ancient Athenian and modern American democracy contain a shared ideological ethos by which power should rest with the citizens themselves. Furthermore, just as Athenian democracy evolved in the Greek context of tyranny, American democracy originally evolved out of the context of the American Revolution (which itself was understood as a struggle against tyranny). With that in mind, while the checks and safeguards are different, American democracy has its own protections against tyranny via the separation of powers between the various branches of government, each one of which is intended to serve as a check against the other two.