I agree with the basic arguments made in each of these posts, but I would add that it is not exactly true that separation of church and state accompanied the end of the Revolution. The First Amendment only guaranteed that there would be no federal establishment--established churches persisted for some time after the Revolution in some states. Massachusetts, for example, did not rid itself of the established Congregational Church until the 1830s. I offer this only as an interesting fact--I agree with the larger point that the Revolution weakened established churches as sources of political authority.
Beyond what has already been written, I think many people interpreted (and continue to interpret) the Revolution as further evidence that the United States is somehow exceptional, favored by divine Providence.
I would think that the American Revolution, with its emphasis on individual freedom, might have made Americans more likely to accept a less structured religious life than might have existed otherwise. Perhaps we have a greater sense of relying on our own means to reach God, rather than following the dogma laid down by religious "authorities."
One of the main reasons some people came to the colonies was for religious freedom. Now that the United States was a country, those different religions had to work together as one. This made accepting differences, through the separation of church and state.
One impact on religion related to the new ideology of individualism that accompanied the success of the Revolution. When America released itself from patriarchal oversight and became independent, that resulted in a similar release of individuals from religious oversight as more people began to question Christian theology.
I would say that the main impact was to separate the church from the state. Up until that point, essentially all colonies had had their own established, official religons. After the Revolution, this practice ended and there were no longer any official churches. This allowed American religious sects to compete with one another on a much more even basis. It also helped to allow many more sects to arise. When there is not any one official sect, it is much easier for more groups to grow because no one group is favored and helped by the government.