I think people like the fairytale stories provided by shows like this (Pop Idol, X Factor, American Idol, Britain/America's Got Talent), and this is just another example. I don't know if it's the same on American Idol, but nowadays on every series of shows such as the X-factor, each remotely talented person has their own sob-story making the journey to stardom with them, living the dream for "insert recently deceased relative."
It's as much a part of the show as the talent is, it's what makes the people sitting on their sofa's watching the TV keep watching. She wasn't just someone who came on stage and had a nice voice. She was a 40-something virgin (irrelevant to many but somehow we had to know) who had never been kissed, looked like the Ugly Duckling, and proceeded to make everyone think twice about judging people on looks alone (at least for the rest of that day anyway.) As at least one of the judges said iirc, everyone judged her when she walked on stage, but you can bet that the people watching felt pretty good watching that woman sing well. I don't believe she has a world class voice, but she has a good voice, she's an unlikely winner which is usually the kind of person who wins these contests, and people like her because of how badly they treated her when she came on stage.
Comparing her to Paul Potts can't really be done, they have different voices completely (aside from gender), but their stories are quite similar, as Potts was a fat bloke who worked in Phones4U or something and didn't look very talented to most people when he walked on stage. He went on to win, and people remember that. I, however, don't think she'll win. My money's on a young welsh boy with the weirdest name I've ever seen, and a pretty good voice too.