The modern U.S. democratic model bears several strong similarities to that of ancient Greece. The U.S. government is divided into the three branches: legislative, executive and judicial. Likewise, the Athenian democracy of Ancient Greece was divided into the Ekklesia, which was similar to the legislative, or law-making branch, the Boule, which resembled the executive, or law-enforcing branch, and the Dikasteria, a rough equivalent of the judicial branch. This ancient model, though, while serving as a useful example upon which to build future democracies, has been significantly improved upon.
Limits on Participation
In ancient Athens, the most advanced Grecian city state, more than half the population was made up of slaves. A small percentage was foreign residents. Of the 100,000 proper, full citizens of Athens, only about 40,000 were men, and women did not have full democratic rights. This means that while the Grecian democratic model was similar to that of the modern U.S., just under a sixth of the population actually had any say in the democratic process. This social dynamic was also present in the early United States, but eventually slavery was abolished, and women were given full voting rights. Grecian democracy never admitted slaves or women into the democratic process.
Representative vs Direct Democracy
The Ekklesia and the U.S. legislative branch are similar in function, but there is a pretty big difference in the people who made up the ancient legislative body and the modern one. During the 40 annual meetings of the Ekklesia, any of the roughly 40,000 male citizens of Athens was allowed to appear and participate. That's a lot of people vying for floor time. In the U.S. this system has been improved upon with the implementation of a Congressional legislative body made up of elected senators and representatives from each state.
Election vs Lottery
The Boule, or Council of 500, and the Dikasteria also did not have elected leaders in the way that the U.S. does today. Instead, the names of influential Athenian tribesmen, in the case of the Boule, or the names of men over 30, for the Dikasteria, were placed in a pool and randomly selected. This lottery system was intended to prevent outside influence on potential political leaders like the lobbyists and special, private interests seen operating in modern U.S. politics. In the United States, today, the President and his supporting executive staff, who represent the federal executive branch, are nominated and must be elected, or are appointed by already elected officials. The U.S. President has advisors, and at the state level, a variety of other executive officials exist, but ultimately, the primary executive power in the U.S. is the President. Alternatively, in ancient Greece, the Boule was made of 500 men with equal power.
Judges and Police
Certain qualifications had to be met before being added to the pool for selection and service in the Dikasteria, but none really reflected any professional legal experience. Alternatively, in the United States, the judicial branch is made up of judges either appointed by existing officials or, in some states, elected by the people, and while it is not required, some experience in the legal field generally precedes appointment to a judicial position. Also, there were no police or attorneys in ancient Greece. Instead, plaintiffs were essentially arrested by the "demos" or politically active citizens, themselves and were defended or prosecuted by citizens as well. This resulted in the court system being abused for the sake of personal grudges, and so, in modern systems, an objective police force and professional legal representatives exist to prevent such abuses.