Not Just Horror Anymore
That it has vampires that aren't scary. (Or not supposed to be scary, at least - I find Edward pretty scary, personally.)
This criticism comes out of a fairly superficial idea of horror - that horror is a show with a certain type of monstrous character, rather than a show intended to scare you. This mindset leads to the idea that any show that includes a monstrous character traditionally associated with horror has to be horror. And if it's not scary, that's not because it isn't actually a horror story - it's because it's a bad horror story.
But the truth is, most of the modern portrayals of vampires are not in horror settings, and that's not a fundamentally bad thing. It takes some creativity to look at a horror creature and go 'what if we saw it from a different perspective'? If done well, it can be pretty cool.
Currently, most vampire stories fall into two or three genres now. There's the supernatural teen dramas, the supernatural detective stories, and then there's a more general category termed 'urban fantasy', which tends to focus mainly on the internal politics and everyday lives of supernaturals hiding among humans. None of these types need to have vampires, of course, but vampires are probably the most common creatures in these stories.
The thing is that the vampires in these stories are intended to tap into other basic elements of human zeitgeist, rather than the elements we fear. Take supernatural detective stories. These stories are essentially serving the same purpose as superhero stories, but without some of the usual conventions of superhero stories (such as costumes and alter ego names). They're about someone fighting for good, and doing so using some supernatural tools not available to most people. This is an idea that appeals to many people, and if done well, these stories can be pretty amazing.
Supernatural teen dramas are another example. It's a stereotype, but one based in truth - many teenagers want to fit in, but at the same time want to be special. (I'd argue many adults feel that way, too.) Most supernatural teen dramas play into that, by presenting a character that seems normal and manages to fit in (possibly with some struggle), but is nonetheless far from normal in truth. How this character deals with their hidden identity, and how they manage to connect with others, either other supernaturals or normal teens, can be good fodder for some excellent character development. There's also the normal protagonist so often seen in supernatural teen dramas, who longs for something to make them special, and has their longing met in a way they never expected. Even so, these characters are often on the edge of this supernatural world, and there's the constant tension of whether or not they truly belong among these weird and wonderful people they've found.
Personally, I love these kinds of stories. They've always resonated with me, ever since I first picked up Animorphs (a story where a bunch of teenagers and an alien turn into animals to fight an alien invasion). Long before I was diagnosed with high-functioning autism, I instinctively knew I was different, and at the same time, I knew my differences were not obvious to or well-understood by most of the people I met. I didn't know there was anyone else in the world like me, and that made me extremely lonely. So, naturally, when I found stories about people that were different but looked normal, I was immediately a fan.
I think whenever we pigeonhole a certain topic to be limited to a certain genre, we lose something. I mean, look at aliens. Aliens are a common horror movie monster, but they can also be a lovable creature that some kid finds and helps to 'phone home', or a superhero who saves the world and does exciting things most humans will never get the chance to do, or a lost teenager growing up hidden among people who are fundamentally different from him. Why can't the same be true for vampires?