Which had the most far-reaching consequences on American culture: the Enlightenment or the Great Awakening and why?
Just for argument's sake, I'm going to say the Great Awakening was more important in our history. Our religious nature and history as a society, and especially our fundamentalist and Baptist segments of society can trace their lineage and history back to the sentiments of the Great Awakening in the 1730s and 40s. Fire and brimstone speeches are commonplace now, especially in the "Bible Belt" of the South, but it was this religious revolution of sorts that really set the idea in concrete as an American trait. It's not that other countries aren't religious, but our nation seems to be more uniquely devout than many other, older societies, and I think it is because, in large part, to the Great Awakening of the 18th century.
This is an EXCELLENT question simply because it could be answered correctly from several historical perspectives. I offer the following...
The Enlightenment political philosophies suggesting natural rights, consent of the governed, separation of governmental power to prevent abuses to the individual to name a few were gaining support in 17th England. The Glorious Revolution of 1688-89 secured the English Bill of Rights for Englishmen and as British colonization of the Atlantic coast continued into the 18th century those political principles came with them.
The Great Awakening of the 1740's was a response from those who believed religious devotion was losing its zeal, perhaps because of the Enlightenment philosophies. The New England and Middle colonies were swept up by men such as Jonathan Edwards and George Whitefield in order to remind people that business, money, and other earthly desires were corrupting their souls.
Over time the Great Awakening split into different factions and ran out of steam, however the consequences of those factions broke down the old Puritan theocracy, led to greater religious toleration which in turn led to a more secular society. It could be argued that if not for the far reaching consequences of the Great Awakening, America's enlightened founding fathers might have been jailed for heresy.
In many respects it's difficult to completely seperate the two. Although the First Great Awakening was a largely religious movement, begun in England by the Wesleys and George Whitfield, it did help foster a more unified sense of "Americanism" among British colonists in the New World. In addition, it encouraged people to not be afraid of confrontation with religious authorities, and to actively pursue social discussion of whether a particular religious teaching or tradition was correct or helpful to the aims of religion. This led to an increased sense that confronting errors in authoritarian structures was acceptable, leading to the same confrontational stance about political themes. This, of course, increased the acceptance of the idea that political authorities could be challenged if they were "wrong."
On the other hand, would this movement have occured without the ideals of the Enlightenment? Enlightenment philosophers such as Voltaire, Montesquieu and Rosseau espoused concepts which in many ways led directly to the Great Awakening. Thomas Hobbes and John Locke expounded the political theories that underlay the American Revolution and all that led to it. So the question is which is of more weight in the scales of history, one movement or another movement which gave it impetus? This is a question that probably cannot be answered impartially. I would say the Enlightenment was more important, because it is the basis of our political culture and it led to the Great Awakening, which is the basis of our spiritual culture. But someone else might feel differently.