Unfortunately God is often very "important" in US politics, although it should not be. I don't mean that religion is worse than government, and in fact, in the past, there was often little distinction between the two. However, the political and legal systems within the United States are founded on the constitution. That is what gives validity to those systems.
But it is quite often seen that politicians who are religious will often try to convince people that God trumps the constitution and so the constitution can be ignored in such instances. We see this with same sex marriage, with warfare, etc.
Let's start with the constitution itself. What is it? It is a contract between the people and the government, giving said government certain powers and responsibilities. That is why it determines the validity of the political and legal system.
In regard to same sex marriage. The 14th amendment requires that a law treat all citizens alike, and yet, there are many, who due to religious persuasion, wish to prevent same sex couples from obtaining marriage licenses.
Moving on to warfare, religion has and continues to be a major player in the excuse to wage war. Bush often invoked god when justifying his decision to go to war. We can see an example of this in the quote "God told me to end the tyranny in Iraq" (http://www.theguardian.com/world/2005/oct/07/iraq.usa)
Adding another example as supporting evidence, we have the push to teach creationism in biology class. While creationism should be welcome in philosophical discussions in school, it is not scientific and therefore should not be in a science class.
Of course, a discussion on how religion influences politics in America would be want without mentioning Islam. It is an unfortunate reality that there is a group of Islamist extremists who mean us harm. Of course, part of that problem arose from secular mistakes, such as funding what would become our main enemies in the middle east during the cold war!.
Lastly there is the counter-religious movement which hopes to remove all religious expression from public spaces. This is often justified by the idea of separation of church and state: a phrase which does not exist in the constitution!. The constitution discusses the free expression of religion. Therefore the argument of whether or not we can express religion in public venues is a large part of political discourse in America today.