SCOTUSblog discusses the case:
The boundaries of where religious freedom ends and anti-gay discrimination begins is the next frontier in the battle over LGBT equality. This photography case has been a cause célèbre among religious extremists and heterosexual supremacists who oppose marriage equality; they make the claim that allowing marriage equality without an exemption for religious people to not participate in providing services or recognizing marriages that they object to tramples their religious freedom. Funny how they don't make that same claim about interracial marriage now even though there was significant opposition to interracial marriage that was based in religious belief.
Proponents for LGBT equality like myself see this claim for religious exemptions as thinly veiled attempts to exempt religious people from the basic tenets of public accommodations law.
In the three huge legislative victories for LGBT people last week (Illinois, ENDA and Hawaii) the extent of religious exemptions was a key point of debate, and this argument is not going away as religious homophobes realize they are ultimately going to lose the fight to stop marriage equality and LGBT non-discrimination nationwide so they want to limit its effect as much as possible.
We shall be watching this case very carefully to see if the U.S. Supreme Court accepts it for review. It should be noted that the New Mexico Supreme Court ruling being appealed was unanimous in favor of the public accommodations law and other cases where LGBT anti-discrimination and religious belief have all gone in favor of anti-discrimination. However, in cases where LGBT rights and other first amendment fights have been in conflict have generally gone against us (Boy Scouts of America v. Dale and Hurley v. Irish-American Gay, Lesbian and Bisexual Group of Boston).
Hat/tip to Joe.My.God