Mutants are feared more than superheroes for the same reason gay men are more feared than lesbians.
Because bigotry doesn't make sense.
The average person doesn't worry about what will happen to them if they are bitten by a radioactive spider....but they do worry about what will happen if their child is born different from everyone else.
These are great points made earlier in the thread. I would also like to add some other thoughts. First of all the amount of mutants compared to the superheros, also add to that, there is only one Spider-Man, only one Iron Man and so on, the reasons to their powers are quite unique in the Marvel Universe. While new mutants are probably born everyday, which adds to the fear that they could be anyone on the street cause most of them look normal. And for those who doesn't look normal, there is of course the freak thing that people fear and get put off/intimidated by.
And it's not like the mutant hate boils down to everyone hating every single mutant. It's probably more the concept of it all, and wanting it under control. Why some people can accept let's say Beast as an Avenger or as a politician and still be wary of the growing mutant influence in the world. It's not just black and white really.
And it's not like other different and powered beings in Marvel Universe haven't met their fair share of bigotry, misunderstanding and fear. And during the Days of Future Past story-line, in the future the Sentinels started to kill not only mutants but other superheroes too, because they saw them as similar threaths.
So often mutants gets to be the group that is easy to blame things on in debates and such. It's also the PR thing as others mentioned, to that also adds that Charles chooses to take in those that might need guidance or feel lost, they choose to be the outcast superhero-team. That way they can represent something else than let's say the more up-front and government controlled The Avengers.
In other words, the X-Men and mutants were created to be a metaphor for outcasts/being different and to highlight the bigotry and hate in our world.