There's no appreciable difference in reasoning between either method. Both require some kind of inherent logical consistency, both require a simple “law” of causality that makes QED after a proof a logical consistent statement, both extensively use empirical evidence as well as inductive and deductive logic and both are specifically designed to explain the empirical evidence. The difference is in the predictions that they make. Science purely limits itself to reasoning about empirical evidence, i.e. this bridge design will support that much weight. Religion however goes two or three steps further and offers damnation and salvation based on essentially the equivalent of nothing but pure extrapolation of previous theories. So that's theory a assuming theory b assuming theory c explaining some empirical evidence. Example:

After praying to $deity some person is cured. Religion first theorises that praying to $deity will work for curing, then goes on to theorise the existence of $deity and finally theorises that $deity has the “power”/“ability” to cure. After that we take a leap of faith (litteraly) and jump to the concluding theory that $deity may be able to grant you an after life (i.e. the ultimate cure, the cure of death...).

Arguing for a separation of “science” and “state” is useless, since it effectively asks for a separation of “reasoning” and state. Arguing for separation of “religion” and “state” is not quite so useless because all it does is restrict us to empirical evidence.

Of course historically this arose for very different reasons: religion has a tendency to have its followers brutally slaughter those who do not follow it and generally interfering with the personal freedom of non-believers. There's as much empirical evidence to suggest God exists as there is to suggest that all religions are inherently violent. But both are a leap of faith and a jump to conclusions based on other theories, for there is plenty of countering evidence which directly contradicts the theories on which these statements are founded.