As I understand Az's position, the crux of the difference lies in that when you come down to brass tacks, race, ancestry, religion, and culture do not create the sort of fundamental difference that exists between Wizard and Muggle.
A child of one (human) race and culture can be taken in as a baby and brought up in another culture with a family of another (human) race, and other than some cosmetic differences and a need to keep an eye out for any health issues common to the kid's biological family, there really isn't that much of a difference. Furthermore, parents of one race cannot engender a child of another race. Heredity is pretty straightforward, and while some interesting hidden recessives may show up and surprise everyone, you can't be, say, part Irish unless one of your parents was also either Irish or part-Irish.
Wizardry, on the other hand, is a relatively clear case of have and have-not between the Wizarding and Muggle population. Wizardry is a pan-racial talent, and either you're a wizard or you're not, regardless of ancestry. If you have the power of possession and control of magic, you're a wizard, whether the people around you like it or not. Unlike an Irish child being born to parents with no Irish blood in them, a wizard can be born to Muggle parents, and wizard parents can wind up with a squib in their crib.
JKR hasn't gone into dreadful detail about how, exactly, magic or lack thereof, is inherited. Likelihood is that it's a gene or a combination of genes doing wacked-out things, or it could be triggered by environmental conditions in the womb, or, who knows, the little magic-fairy might go around kissing children and blessing (or cursing) them with magic. Until she tells us, we can speculate all day long, but we still won't know.
Az's position, as I understand it, is that since heredity of magic and heredity of race work differently, and since there are real root-level differences between wizards and Muggles, and there really aren't that many differences between different human races on that scale, saying that anti-Muggle, anti-Muggleborn, pro-Pureblood prejudice is equivalent to racism is logically flawed.
I disagree with the position because I believe that people who hold racism on as extreme a level as Bad Tommy's gang of thugs hold their blood prejudices truly believe that there are differences as extreme as the differences between wizard and Muggle between their hated race and their preferred race. The difference between races may be an illusory other than the cosmetic and the cultural components, but not as far as hardcore racists are concerned. Furthermore, I think that a whole lot of the core members of the Death Eaters are, in fact, so insular and Old-Vor about bloodlines, breeding, and having had magic in the family since time immemorial, that they are in fact taking their prejudice beyond magic vs. non-magic.
One of the rational arguments I heard in favor of the old wizarding families' blood prejudices was, IIRC, in Pawn to Queen, where two of the Slytherins told Hermione that Muggleborn wizards and witches should marry other Muggleborns for as many generations as it took in order to make sure that the magic bred true before introducing their genes into the wizarding society at large. That's a reasonable idea, purely from the viewpoint of wanting to strengthen the genetic tendency towards magic.
But I don't think that half the people who believe in the superiority of wizardkind over Muggles is that rational about the topic. If only all of them were, and did base their opinions firmly on the idea that wizards and Muggles are different, and the magic genes will eventually start breeding true, and stopped wanking over their family trees, then I'd agree with ataniell93 that the analogy between Pureblood-superiorists and racists is flawed.
As it is, since Muggle racists can't grasp hold of the fact that there's really no difference between Us and Them, and certain wizards have trouble letting go of the fact that once someone's bloodline has started breeding mostly wizards, that they're an Us and not a Them, I still say that racism and the Muggle/wizard conflict have a lot in common.