Although both the United States and ancient Athens are described as democracies, the systems of government are quite different.
Since the advent of women's suffrage in the early 20th century, all citizens of the US over 18 who have not committed felonies are able to vote. In Athens, women, slaves, metics (resident aliens), and those who did not have Athenian grandparents could not vote.
Another salient difference is that Athens was a democracy and the United States is not. It is a Republic, modelled on republican Rome. Citizens (accept in certain states, such as California) generally do not vote directly for laws but instead vote for representatives to legislative assemblies such as Congress; those representatives vote for laws. Athens was a direct democracy in which citizens were part of the assembly and voted and spoke for themselves rather than through representatives.
Finally, for many important positions, people in Athens were selected by lot -- a blindfolded priestess from Delphi would reach into a urn containing potshards each with a name -- to avoid corruption, whereas selection by lot is rare in the US (except for limited use in jury selection)