Well i don't believe that Rome and America can be compared like banana's and plantains, but a lot of undercurrents in our society are from the Romans. I must also point out that Rome as a city and rome as an empire (as well as the byzantines) look very different from one another, although they are technically part of the same thing. America and Rome are technically part of the same thing, although what we inherited are more echoes through British culture than any direct borrowing.
What i mean is, the histories of the three entities are interlinked and in some ways similar. When the Roman's entered Britain, they were confronted with technologically inferior tribes. They were Roman for only a small chunk of their history (albeit the one that formed the foundation for the modern state of Britain. The creation of London, and so on). Britain (as well as the Roman emprie) was transformed by an infusion of Germans (eventually ending in the technical end of Roman power). Despite the Roman's loosing their grip on the Isle, there was still a Romanesque culture. This was changed dramatically but not eradicated by the Anglo Saxons, Normans, etc. Now once Britain becomes a great power (by defeated other Roman/German descended states, France and Spain) they were free to expand Westward, as the roman's did. They spent roughly a century colonizing the east coast, which eventually broke away. They, like the Romans, ceded control rather than fight a difficult battle. Eventually the British Empire waned, and America had formed its own. We are an expansionist state that generally seeks to draw our subjects into our culture.
When i wrote my first post, i was a bit hurried, so i just made bullet points. But i really do believe we are a direct, if not very mutated descended of the Romans. They radiated outwards, creating the British on the fringes, who radiated outwards, and created America on the fringes. Also, it is important to note that there is a significant gap between the creation of the state and its global dominance. The thirteen colonies were "America" in the sense that they are part of the same entity that exists today, but are at the same time barely recognizable. All three are vastly diverse (although, at the same time, retain a degree of homogeneity). All three are christian, but also changed what it means to be Christian.
I also cannot stress enough how the English we speak today is a direct result of the Romans. While it is Germanic, there has been a massive infusion from French (and we are perhaps beginning to see a mixture with spanish.) which added to an already large number of words rooted in Latin.
I could go on and on. I'm hoping that someday i will. I hope you all, before picking this all apart, realize i'm making some broad generalizations. There are many other factors that went into the creation of these two states. But i think that historians in the distant future will link the three of us together, because the cores of our culture remain the same. Perhaps it is difficult to see because of the point we are in history. Once America's time is over, we will look the same. I also must say that i'm the type of person who, when they look at history, thinks nothing ever ends. If you turn on the news and see a story about the Tea parties anger at the IRS, it is because we have been fighting the same battle throughout our entire history, just manifested in different ways.
PS. one last thing i forgot to mention is genetics. The British are genetically only a tiny fraction "Roman." Less than ten percent of American's have British heritage. Far more of it is German