What is liberty of conscience?

I have a question for you, basically about terminology. Here is a quoted passage, which I have never understood.

Should religious organizations and their members be treated as unequal under the law for certain purposes connected with gender? US constitutional law has standardly granted special latitude to religion, by contrast with other forms of commitment and affiliation. Religious reasons for exemption from military service, or for refusing to work on a particular day, are granted a latitude that is not granted to other forms of conscientious commitment, such as the familial or the artistic or even the ethical. This remains controversial for the way it appears to privilege religion over nonreligion and thus, it might seem, to violate the Establishment Clause…[S]uch privileges given to religion, though highly contestable, can be strongly supported by pointing to the special importance of liberty of conscience as a fundamental right and the consequent need to give religious freedom special protection…

It may be obvious what I don’t understand. If liberty of conscience has special importance, why is that a reason for privileging religion while not privileging other forms of conscientious commitment? They all have to do with liberty of conscience, right? If so, why is liberty of conscience in the concluding sentence taken to refer to religion and not other forms of conscientious commitment? Perhaps I’m wrong and liberty of conscience actually does refer only to religion – but if so, why? Surely ‘conscience’ doesn’t mean ‘religious conscience’ so why would liberty of conscience refer only to religion? Help me out here.

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