THEY DO! Shortly after the Revolutionary War, Upper Class British started using non-rhotic speech (they drop their R's), and it slowly trickled into the speech patterns of every day citizens. The British accent as we know it now is simply different now than it was then. The Boston accent is a good example of non-rhotic. Try saying "Park the car in Harvard yard" in a Boston accent and then again in a British accent, you'll hear the similarities. The more nasal and clipped way we speak is probably due to German influence, add in Native American Speech patterns, plus a wave of immigrants from all over the world, and new American accent was born, as well as a whole lot of regional ones!
