The Constitution was written by Christians, many of whom were strong believers in their faith. The Constitution specifically separated Church and State in order to avoid two things: the State establishing a central Church, and any religion from seizing control of the state (which would lead to the establishment of a central Church). Neither of them is more important than the other because they lean on each other.
The separation of Church and State was a response to the taxes the colonists were forced to pay to the Church of England. As such, the State cannot sanction a religion and make policies based on religious beliefs. The State is meant to be run free of religious views so that religious discrimination or discrimination "justified" by religion would not have grounds in legislation or execution of the law.
Religion is a deeply polarizing subject, and has either caused or been used to justify some of the worst wars, atrocities, and other horrific acts throughout the ages. For example, the Crusades, the Buddhist Vietcong in Vietnam, the Inquisition, the rise of Islamist Terrorism and ISIS, and (arguably) the Holocaust. These are examples of what may happen when religion and government are too closely interwoven in a state.
The other side is that the State cannot interfere with the workings of the Church. The State cannot regulate what the Church teaches or preaches, who may attend, where and when sermons or services may be held, etc. The State is meant to have a Hands Off policy to prevent the abuse of power against members of a non-State-sanctioned religion.
Communism is an example of the State meddling in the Church's affairs. Religious bans or limitations cause tension between religious organizations and the State, which doesn't often end well for either.
One of the main ideals of the separation of Church and State is that the government takes care of the economic and social needs of its citizens, while religion tends to their followers' spiritual and moral needs. On their own, corruption will eventually sprout, but when combined the corruption goes from being one or two ugly plants in your yard to BEING your new yard. We are seeing this happening now, albeit at a much slower rate than during the dark ages, when the Catholic Church had a stranglehold on spiritual AND economic and social authority.
Basically, what I'm saying is that I agree with the second part of SpursFan's statement, but not the first. Neither should meddle with the other.