Both books deal with the notion of heredity and ancestry, but they treat it very differently. In the Twilight books, fate is determined by birthright. In the Potter books, birthright is presented as purely a social construct designed to oppress. Think Jacob and the Native American werewolves in Twilight, doomed to their fate through blood ties, versus Hermione and the other 'half bloods' or children of 'muggles' at Hogwarts, who are continually shown to be deserving of their place at the school in spite of elitist snobbery from Malfoy et al.
No surprise, then, that Rowling herself is a progressive. She was a single mother when she wrote the first Potter book, living on welfare. Now estimated to be worth $1.1 billion, she gives massively to progressive causes the world over. No surprise, either, that her books terrify Christian fundamentalists. Potter books have been banned by many far-right Christian groups.
No surprise, either, that Meyer is a devout Mormon, a graduate of Brigham Young University who says on her Web site that her religion colors everything she writes. She describes herself as "very religious," and her series ends with the female protagonist, all of 18, marrying Edward, becoming a vampire, and bearing his monster child.
There are many examples of Mormon theology flooding Meyer's work, some of it racist against Native Americans, Latin Americans and anyone with dark skin, much of it sexist in the sense that Bella does not exist but to love Edward. (Meyer's adult novel, The Host, is essentially a retelling of the Book of Mormon, set against a sci-fi backdrop.) The constant criticism the Twilight books have received is that Bella is not much of a character; there is no core to her, other than her adjective-laden obsession with the vampire.
Below: "Conservatives understand that we minorities have dark animal sides! Heck, I would've voted for McCain if I'd been old enough!"