This is a very difficult question to answer since both of these movements have had a great deal of impact on American society. However, I would argue that the Enlightenment has had more of an impact.
The Great Awakening has had a tremendous impact on American society. Arguably, every evangelical Christian today traces their religious convictions back to the Great Awakening. This was a movement that promoted the idea of a personal relationship with God. In this way, the Great Awakening led to the existence of one half of our “culture wars” today.
However, the Enlightenment probably had a greater impact. First, it is the other half of the culture wars. It is the side that maintains that we should rely more on rationality than on religious faith. More importantly, though, it can be argued that our entire society is based on Enlightenment ideas. Even most evangelical Christians do not argue with the idea of the separation of church and state, which is an Enlightenment idea. Most people believe in the scientific method (though there are a shocking number of people who apparently do not believe in things like evolution). Finally, our entire system of government is based on the Enlightenment. John Locke, whose ideas are behind the Declaration of Independence and the Constitution, was a major Enlightenment figure.
Thus, while the Great Awakening has had a lot of impact on our society, I think that the Enlightenment has had more.