The most telling similarity between Ancient Greece and modern day America is the idea of democracy practiced both then and now. Both Ancient Greece and the present-day United States practice democracy. The idea behind them are the same, but they differ on practical considerations.
In Ancient Greece, the form of democracy practiced is known as a direct democracy. This form takes the idea "of the people, by the people, and for the people" as literally as possible. The people themselves could have their voices heard by the government when they had concerns they wanted to express. There was no mediation between the common person and the government.
In present-day America, however, representative democracy has replaced direct democracy. Representative democracy is not diametrically opposed to what direct democracy sought to accomplish. In a representative democracy, rather than have the common person airing their grievances directly to the government body, a representative who stands in for a portion of the population expresses the concerns of their constituency.
While this seems like a starkly different system than that practiced in Ancient Greece, the primary difference between them is the scale of the population. A representative democracy has taken the place of a direct democracy in America primarily because of the population involved. Since the voting population in the United States vastly outnumbers anything Ancient Greek society could possibly imagine, it is practically impossible for every member of the voting population to directly address the government. For this reason, those individual common people in Ancient Greece have become the individual representatives in present-day America. Representatives in American politics, like the common person in Ancient Greek politics, represent the common interest.
In Ancient Egypt, the belief system upheld by the priestly class took a dominant role in Egyptian society. One of the most important beliefs in Ancient Egyptian religion was the idea of an afterlife. When anyone was buried, whether it was a pharaoh or a commoner, all believed that they would exist in an afterlife. In addition, anything from floods, to famines, to productive harvests, correlated with the will of certain gods. The direct connection between the realm of the gods and the realm of human existence was unmistakeable. On an individual level, a bout of bad luck was usually construed as a falling out of favor with the gods. For these reasons, social practices among the Ancient Egyptians were geared toward achieving circumstances amenable to divine favor, so that society would flourish.