There are two main elements in what has been called religion in the West for the last few hundred years: on the one hand it is private – the basic place of true religion is the heart and conscience where the individual communes with God. On the other hand, religion is seen as containing the threat of irrational violence. It deals with absolute values and is based on faith, and so gives warrant for extreme actions. From the Thirty Years War to 9-11, (so the argument goes) there is a reason to see religious disputes as particularly irrational, irresolvable, and extreme.
The necessary consequence of both these ideas is that the public as such must be irreligious and is charged with making sure that religion is kept private and away from the coercive tools of power. Thus the separation of Church and state is defended both by the religious, who are convinced that this is necessary to preserve the private, poor, and devotional character of true religion; and from those who are hostile to religion and see it as a source of fundamentalist extremism. Thus both sides work for the same mediate goal, but have completely opposite ultimate goals: the religious divide Church and state with the hope of making the Church more pure and more perfect; those hostile to religion divide Church from state as a half-measure to eliminating it altogether.