The most obvious example of how religion and the Enlightenment changed America in the 1700s is that of the Revolutionary War. The American Revolutionary War was the war against Britain in which America obtained its independence.
This war was justified by the most significant and powerful religious group in the thirteen colonies: Puritans. Puritanism argues that society is bound together by a social covenant. Once the people have entered into such an agreement, voters were responsible for choosing qualified men to govern them. If the ruler was evil, they had the right and duty to oppose him. This way of thinking meant they had also supported some other revolutions in the past, such as the Glorious Revolution of 1688.
The Enlightenment also played a big role in the Revolutionary War and its aftermath, particularly with the Founding Fathers. Being all advocates of the Enlightenment's ideals, they built the new Constitution and the new government around these values. Therefore, things that are intrinsic now to the political process, such as freedom of speech, men as having inalienable rights and all citizens being equal under the law, are product of this movement.