Why religious exemptions?
The most famous issue on which the Catholic church claims an exemption is the law requiring health insurance to provide birth control coverage. The logic used here is the classic "conscience" logic. To put it in utilitarian terms, being forced to do something that you believe is deeply wrong is harmful to a person, and may be more harmful than the consequences of allowing the person to avoid the requirement. The suffering of a Quaker forced to join the army is greater than the harm to national security of one less set of boots on the ground, and so we grant them conscientious objector status. Conscience logic is fundamentally secular, as it's a function purely of the depth and sincerity of one's belief, regardless of the correctness of those beliefs (though in practice religious consciences are more likely to succeed in their claims).
The explicit argument in the birth control case is conscience-based. The church argues that covering birth control for its employees* would be a grave violation of the Catholic conscience. After all, opposition to non-procreative sex is the central pillar of modern Christianity. The debate is then about whether the harm of allowing that exception (reducing access to birth control) is great enough to outweigh this harm to conscience.
The logic employed in Duquesne University's objection to unionization of its adjuncts is quite different. Duquesne claims that as a Catholic institution, it is exempt from National Labor Relations Board rules requiring it to recognize the adjunct union. This case cannot be covered by the conscience logic. Allowing adjuncts to unionize does not violate a fundamental Catholic principle -- not only does unionization have little to do with sex, but the Catholic church has staked out an official position explicitly in favor of unionization. Instead, we can call this a "separate spheres" logic. Here, the claim is that the Catholic church constitutes a distinct institution that should have broad discretion to run its own affairs. Unlike in conscience cases, the church would not have to prove the validity of its policies to the secular government, any more than Pennsylvania has to justify its own internal laws to New York.
Separate spheres is, however, a logic that requires state consideration of the validity of religion. The government has to decide which institutions deserve to have separate spheres apart from the general law. These separate spheres need not be religious -- similar separate spheres are granted to Native American tribes, for example. But if an institution wants a separate sphere, it must justify that, a justification that for the Catholic Church would have to be religious in nature.
The separate spheres logic is not unique to the Duquesne adjunct case -- the church has used it, for example, in trying to shelter priests accused of molestation or molestation-enabling. And it has the capacity to subsume conscience cases, for example by arguing that as a separate sphere, the church is free to make its own decisions about health insurance coverage on any grounds it pleases.
*Wording adjusted from original version