What happens in the public place, and how does it affect Romeo and Juliet's plans?
The prince has expressly forbidden fighting in the street, on the pain of death, so fighting alone could have cost Romeo his life. When Romeo kills Juliet's cousin, Tybalt, he is given a light sentence: exile.
This does interfere some with the newlywed's plans. Both are young and neither have the means to life well on their own, so when Romeo is exiled, Juliet stays.
It would be gross understatement to say that the marriage is in trouble before it is even consumated.
By the time the public brawl is done, Romoe's reputation is ruined and his new mother-in-law wants him dead; to make matters worse, his new father-in-law wants to ease Juliet's pain by marrying her off to someone else. Neither of them have time to tell their parents of their marriage, so neither of the Capulets know. The nurse--one of the few people who does know.(Of course, with Juliet's need for secrecy, one wonders when, if ever they would have told her parents). The nurse, one of the few people who do know, encourages Juliet to accept the new (and better) match with Paris. This sparks desperation in Juliet, who feels trapped and alone. And desperate minds are too willing to take desperate measures.
In Act III, it is midday just after the private wedding of Romeo and Juliet. Mercutio and Benvolio are walking trhough the square when they meet Tybalt who asks about Romeo, whom he has challenged to a duel for showing up at the Capulet's ball. Insults are exchanged leading to Mercutio's death. Romeo then kills Tybalt in revenge. The prince comes to the square to investigate the fight and ends up exiling Romeo. The exile forces Romeo to leave Verona and Juliet putting into motion the final tragic acts of the play.
Act III is the climax and turning point for most of Shakespeare's tragedies including Romeo and Juliet. The actions that take place in the public square in Act III provide the turning point for the romance between the two characters who have been hiding their romance contrasting the private versus public nature of the love and hate in the play.