The difference between how the two of them see the trial going can be seen quite nicely in the following two quotes from the book:

Jem seemed to be having a quiet fit. He was pounding the balcony rail softly, and once he whispered, "We've got him."

and

He could easily have done it. I thought Jem was counting his chickens.

What is going on here is that Jem actually understands where Atticus is going and Scout does not.  We have not yet seen that Tom Robinson's left hand is pretty much worthless.

Ironically, though, Scout is the one who is right about how the trial will turn out, even if she is wrong about what is going on right at the end of Chapter 17.  Jem is right -- Atticus has made a huge point.  But Scout's view is correct for the trial as a whole.