How is Jem changing, and why do these changes bother Scout?

Simply put--Jem is growing up. Calpurnia tries to explain it to Scout in chapter 12 as follows:

"Baby. . . I just can't help it if Mister Jem's growin' up. He's gonna want to be off to himself a lot now, doin' whatever boys do, so you just come right on in the kitchen when you feel lonesome" (115).

By chapter 14 the most significant changes start to show in Jem. He starts to act more like an adult to Scout rather than treating her like an equal or a playmate. For example, he takes Scout aside and tells her that if she doesn't stop aggravating Aunt Alexandra, he will spank her. Scout gets into a physical fight with him about this, and since he fights back, she feels he hasn't changed enough for them not to still be equals. But the change in Jem certainly strains his relationship with Scout.

Following this fight, Scout discovers Dill hiding under her bed. He had run away from home and she wasn't going to tell Atticus about him if it weren't for Jem. When Jem finds Dill, he says, "You oughta let your mother know where you are" (141). Then he goes to get Atticus to tell him that Dill is in Scout's room and she says the following:

"Dill's eyes flickered at Jem, and Jem looked at the floor. Then he rose and broke the remaining code of our childhood. He went out of the room and down the hall. 'Atticus,' his voice was distant. 'can you come here a minute, sir?'" (141).

Jem breaks the code of no snitching on other kids. He acts like an adult by telling on Dill and it almost makes Dill run again. If it weren't for Atticus's calm and kind response, he may have run right then. Fortunately, Jem and Scout figure a way to stay brother and sister as Jem goes through puberty and matures--just like all siblings do.