How does Kim Davis' conduct differ from Theocracy?

As Kim Davis sits in jail on Contempt of Court charges for failing to perform her duties as county clerk or allowing her deputies to perform those duties for her, there have been claims of religious persecution.
Mike Huckabee has stated that her jailing, “removes all doubt of the criminalization of Christianity in our country.”. Many others have made similar claims, yet they have no issues forcing non-Christians to violate their morals.
It seems to me that when one allows religion to trump secular law, that we cease being anything other than a religious state where clergy (of one faith) wields all power and deprives all non-followers of the privileges that they themselves consider God-given rights.
Sure, it’s the Bible instead of the Quran and Allah instead of God, but those are mere details. Can you tell me how Davis’ conduct and the support of such behavior differs in any practical way from Sharia Law or other blatant theocracies?