Jem's reactions are extremely emotional and centered on injustice. Scout is more blatant, matter-of-fact, and in your face about what goes on.
Jem cried and remained quiet and moody for quite some time. I think at 12 years old or so, Jem saw the injustice for what it was and lost a little faith in adults to do the right thing. That is the disappointing truth about adulthood, sometimes we sacrifice the right thing for social acceptance... sound familiar?
Scout was almost too young to understand. She accompanied Dill as the trial and the treatment of Tom Robinson made him physically sick. She treated the situation as if that's how all the trials are. This shows her immaturity and inability to understand the complexity of the issues presented, but nonetheless, her innocent narration provided the details in a straight forward manner that totally makes the audience aware of the severity of race relations at the time.
Each of these two experience the concept that you can't understand what people are going through because you can't be someone else. Jem doesn't get the jury until his dad explains it to him; Scout doesn't get Boo until she takes him home and looks at the world through his perspective, literally. Jem's jury lived in a society that would judge them for letting this man off and that would socially cost them. Jem couldn't relate to that. But Atticus could explain it to him. There are things we can understand in our head but not accept in our heart. This is where Jem was.
Their reactions are important because it reminds us that as we age, we need to consider how our actions or prejudices matter. Sometimes we become so accustomed to them that we don't notice how we hurt others. We need to be more like Jem and have concern for all people's rights to be treated fairly.