Our government is of the people and a church is a community of people. So you have the curch and state as the same subset of people. There is going to be overlap. So seperation is nigh impossible as a concept.
Thomas Jefferson made clear that the purpose of the First Amendment was to establish a "wall of separation" between Church and State in order to protect individuals' right of conscience. So it was to protect the religion from government.
It is important as a society to have limits on behavior. It is important that a State has limits on behavior as well. When it comes to religion the limits are set up (usually) by preternatural authority. Example being the 10 commandments. or the 613 Mitzva of the Torah. or the 1500+ Fatwa of Islam. So Divinely inspired and humanly inforced by people whom are both participants in religion and state.
So what happens when a limit on behavior by the state is in direct conflict with a limit on behavior in a religion? Anything from debate to war will occur.