The first example that comes to mind is in chapter two when Scout starts school. Jem gives her strict instructions to stay away from him at school, and to not talk to him about anything, especially their games. This is quite a change for Scout, whose sole entertainment and companion her entire life has been Jem. School itself drives them apart a bit.
In chapter five, Scout mentions how Dill and Jem "spent days together in the treehouse plotting and planning," and leaving her out of their activities most of the time. So, there is one example of Scout and Jem growing apart. Scout spent the time missed with them "sitting with Miss Maudie Atkinson on her front porch."
Later, after the escapade where Jem lost his pants, he is telling Scout he's going to go get them back. She doesn't want him to, and he is afraid that Atticus will find out and give him a whipping. He says after this, "We shouldn'a done that tonight, Scout." His declaration of guilt over sneaking into Boo's yard and then lying about it is bewildering to Scout, and she says that it was then "that Jem and I first began to part company." What she means there is that she doesn't understand why Jem would feel bad about getting away with something, and risk getting caught to get his pants. She is just relieved that she didn't get caught. Jem is developing a conscience, a feel for things outside himself, whereas she is still in her own head, relieved to not be caught. So, they grow apart a bit there too.
A fourth example is when Jem turns twelve. He becomes "difficult to live with, inconsistent, moody" and tells Scout to start "bein' a girl and acting right!" This is very upsetting to Scout, and she burst into tears and runs to Cal for comfort. So, Jem's hitting the age of 12 is an event that puts some distance between the two.
I hope those examples help! Good luck!