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Things that are not like Kim Davis:

A person being drafted involuntarily into the military asserting conscientious objections to serving in the military.

The difference: the word “involuntarily.” Davis’s situation is like a pacifist enlisting in the all-volunteer army and then asserting conscientious objections to serving in the military.

It is illegal to discriminate on the basis of religion, 42 U.S.C. 2000e-2(a), which is defined as

all aspects of religious observance and practice, as well as belief, unless an employer demonstrates that he is unable to reasonably accommodate to an employee’s or prospective employee’s religious observance or practice without undue hardship on the conduct of the employer’s business.

42 U.S.C.A. § 2000e(j) (emphasis added). So if you’re a county clerk and you want to wear a yarmulke or headscarf to work, you can because it will not affect the conduct of the employer’s business. If you are a pharmacist, the whole point of the employer’s business is to sell medicine, so you cannot require your employer to accommodate your Christian Scientist beliefs by removing one of your primary job duties.

Kim Davis is not a conscientious objector; she’s a person refusing to do her job or possibly a person who has the wrong job.

Update:  Eric commented

One difference: when Kim Davis took her job, issuing gay marriage licenses was NOT part of her job.

My easy and, I think, correct response was that issuing marriage licenses was part of the job.  That should be the end of the discussion.  But it made me think:  how would I handle Kim Davis’s case if it were a real case — as opposed to a publicity stunt — and she came to me for legal advice?  Distasteful as I find her religious views, I would advocate a role for her that did not require her to personally issue marriage licenses to people she did not approve of.*  There appear to be a number of employees in the clerk’s office willing to issue all legal licenses; if the office could permit her to refuse to issue such licenses “without undue hardship on the conduct of the employer’s business,” I think that would be permitted.  The distinction is like a Wal-Mart employee who won’t sell guns — which should be permitted as it’s an enormous store with plenty of other tasks — and a gun store employee who won’t sell guns — which will cause undue hardship as the employer would be paying the employee to do nothing.

The problem with this analysis for Davis is that she apparently instructed the rest of her staff not to issue marriage licenses to same sex couples, which did cause hardship in that the entire office ceased to be able to do one of the things it is tasked with doing.

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*The test for religious accommodation requires sincerity but sadly not consistency, so she can decide to disapprove of gay and lesbian couples, while continuing to issue licenses to people of other (and different) faiths, divorce(e)s, child abusers, tax cheats, lusters, gluttons, sloths, greedy bastards and other sinners, deadly or otherwise.