The first character to show bravery against a mob is Jem. When men begin to move dangerously close to Atticus, Jem breaks the tension when he tells Atticus the phone is ringing. This allows Atticus a graceful way of escaping the mob without showing cowardice. Scout's bravery occurs in front of the mob outside the jail where Tom Robinson is being kept. She breaks the tension by singling out Walter Cunningham, member of the mob, and talking to him about his child. Embarrassed by his actions, Cunningham orders to the mob to leave. Both of these acts of bravery have been proceeded by Atticus' questions which lead the children to really think about what they believe. Their actions show they have learned what to believe and the importance of taking a stance on those beliefs.
I'm not entirely such which two mobs you're talking about, but I know one of them is probably the mob in front of the jail when Atticus is protecting Tom Robinson.
There's a slight difference in the lessons that Jem and Scout learn and the lessons the audience should take away. I would argue that Scout walks away from that situation without realizing how much was going on. Jem, being the older brother, realizes the danger and learns how brave he can be when protecting those he loves. He refuses to go away even though Atticus orders him to do so. Jem stands up to his father and he (and by association, Scout) protect their father and Tom Robinson.
It would seem that Jem also walks away with the knowledge of what his father is willing to do and put on the line for this serious case.
The reader should learn some about mob behavior--how dangerous it can be and what terrible things can happen when people don't think for themselves (which is a theme throughout the whole novel.) The lesson of putting yourself in someone else's shoes is shown here when Scout, mentioning Mr. Cunningham's son, puts MR. Cunningham in Atticus' shoes--Mr. C has kids as well; would he want to kill/harm these children's father?