@strawCake - I think you're right about the police and fire departments. I think it's important to note, though, that off duty police officers and firefighters have to follow the rules of the road just like everyone else!
Anyway, I was pondering what I read in the article and I think one big different between general law and private law are the possible penalties for breaking those laws. If you break a general law, you are prosecuted by the state and you can go to jail or have to pay the government a fine.
In contrast, when you break a private law you're not usually prosecuted by the state. Instead, you can be sued by the other private citizen you've wronged by breaking the law. If you lose the case, you may have to pay damages to the person you've wronged, but you won't go to jail.