You're half-wrong there.
Paedophiles are generally not born paedophiles. There are indeed psychological and environmental causes for paedophilia (including being molested as a child)
That being said, it's generally not something someone chooses to be. Making paedophilia illegal would be akin to thought police.
To sum up the major difference...
Paedophilia is a state of being, child molestation/rape is an act.
Just like zoophilia is a state of being, and bestiality is an act.