"attracted to children specifically is different only in the level of younger, but you're over-generalizing here."
No. It's a level of emotional maturity. As in can or cannot handle an adult relationship and all of its responsibilities. The state puts a general number on it of 18 (or so, depending on the state and country), just like for a driver's license. And just like a driver's license, there are kids who are responsible enough before 16, and there are ones who aren't after 16; the state just picks a *cough* decent *cough* number.
It's all about adult responsibility, though, where true pedophilia is about being attracted to children who *aren't* mature enough for that responsibility. That's why someone 50 can be psychologically healthy and be attracted to someone half their age, while someone 24 is not healthy being attracted to someone half their age.
The biological part you neglect to mention is that they seek someone younger - but still has the traits of being able to produce children. ie, after puberty. Pedophilia is (psychologically, not legally) about the attraction to humans before (during?) puberty.