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.
Scout means that this is the point where she realises that her and Jem are no longer as comparable as they used to be- Jem was reaching puberty and maturing, no longer needing the presence of his little sister as often as before. This is the point where Scout decides their parting of ways begins. This happens becasue the reason that 'troutmiller' gave below.
Th quote is actually at the top of page 63, if you have the black book with the orange mockingbird on the front, it depends on which published version you have.
At this point she recognizes that her and Jem are not as alike as she thought they were and as Jem is maturing into a young man, he does not always want to take part in Scout's excursions. While Scout is very scared, Jem is viewing things in a more adult manner and believes he will be alright.