I think the most threatening part of the zombie apocalypse story is man. I'm surprised more people haven't put this down. The zombie apocalypse strips away the facade of personality that a cozy life affords us and shows a person's true colors. Like Harry, from Night of the Living Dead, who refuses to help Ben reinforce the ground floor and instead goes into the basement (where he and his family die). The inability for both Ben and Harry to comprimise leads to their deaths. This can go on forever, like in Land where the zombie has become a plaything for the survivors, or in the Dawn remake where they play a game consisting of finding zombies who look like famous people and watching the gun store owner, Andy, find and kill them. People selling each other out to live, betraying each other, killing each other.
Ripley from Aliens says it best, "I don't know who's worse, us or them. You don't see them screwing each other over for a half percent." (paraphrase, of course)
I'd be more afraid of betrayal than anything else. I know what the zombies want, what they think. I can't tell what the person next to me thinks. Not even my own brother, or my wife.