I think this is why a lot of people are dismissive of "bullies" and tend to respond along the lines of, "Well, just ignore them; they're just jealous/sad/have a bad home life and they'll find something more entertaining to do after a while". I suspect most people regard "bullying" as sort of a "kid word", whereas "harassment" is an adult word; notably, it's even used as shorthand for sexual harassment, and that's pretty much defined as a grown-up thing: if you act that way toward a kid, it's "molestation", and the connotations are completely different.

I was one of those people until not too many years ago -- to me, "bullying" was what the kids at school did to me, i.e. making fun of me for reading too much or acting horrified because I wore/did/brought something I thought was cool and they thought was stupid. Basically, I was picked on, and that was my mental image for "bullying" -- which meant I tended to dismiss those who got up in arms about it as overreacting.

...All of which is my longwinded way of saying I think you've got a significant point, there -- sometimes having a separate word for a problem just because kids are involved can carry the suggestion that the problem is itself childish, and therefore not something adults need to worry about.