On page 56, Scout says, "It was then, I suppose, that Jem and I first began to part company." Why does this happen and what does it mean?

Jem has to go back and get his pants from the Radley fence.  If he doesn't, he would be caught, and as he said, "Atticus ain't ever whipped me since I can remember.  I wanta keep it that way."

Scout can't fathom him going and getting "killed" by Boo or any other phantom on the Radley property.  She is thinking like a kid who has grown up with superstitions.  Jem, on the other hand, is thinking a little more in an adult manner.  He knows what he has to do, and he knows that he'll be ok.  His pride and his father's approval is far more important to him at this point.  He is going to go.  She can hardly stand the thought of it.  That is where she realizes that they think and act differently because of both the age difference and their level of experiences in life.