This is an intriguing question because Ancient Rome and Greece functioned as ideological models for the establishment of the American Republic. If you think of the architectural style of American buildings where important political decisions are taken, such as the Capitol in Washington DC, or that were residences of important politicians, such as Monticello for Jefferson, you will immediately notice the classical modes inspired by Ancient Rome.
I would argue that one of the most apparent similarities between the two systems of government is republicanism. After being governed by the semi-legendary seven kings, Rome became a Republic in 510 BC with the removal of Tarquinius Superbus and the election, the following year (509 BC) of the first two consuls Brutus and Collatinus. Rome remained a republic until the rise in power of Octavianus, who in 27BC adopted the name of Augustus and inaugurated a series of non elected leaders. The word "republic" itself derives from the Latin "res publica" ("public matter"). Having just attained their independence amidst politcal, economic and social divisions, eighteenth-century Americans looked at Ancient Rome for Republican virtue, the value that could ensure a common ideological reference to overcome their pressing problems. Republican virtue was defined as the (mostly masculine and white) ability to act for the common good and was firmly established in the context of Independence. As Jefferson wrote "Dependence begets subservience", thus linking dependence from Britain to the citizens' inability of maintaining a virtuous conduct.
Yet, this focus on virtue led Ancient Romans and early Americans to limit the political franchise and the right to vote to the people they believed virtuous, that is mostly to wealthy males. This is one obvious difference between Ancient Rome and the present system of government in the USA, where suffrage is universal.