The above answers are correct in citing the main similarity between these two scenes is the fact that these are the two main scenes of physical fighting between the Capulets and the Montagues.
What makes these two scenes most parallel, in my opinion, is the way each fight starts. Though the fight in Act 1 is begun by two servants, it starts with the exchange of petty insults. The fight grows larger because members of each house, without thinking about the reason behind the conflict, join the fight to stand up for their respective families.
The fight scene in Act 3 is very much the same, except that this time, though Tybalt believes he is being insulted by Romeo's kindness, Romeo's intentions (for once) are pure. Tybalt does not even recognize this because he has grown so accustomed to automatic scorn and hatred (both for and from) the Montagues that there is no reason for him to consider Romeo to be genuinely kind.
It was necessary for Shakespeare to include the first fight scene to set the tone for the second and to show the sort of hidden seriousness of the hatred between the families. Both are started because one house feels insulted by the other. Neither of the fights are justified which heightens the pettiness of the overall feud between these families. Additionally, there is a bandwagon effect in both fights, which displays the sense of automatic (if irrational) loyalty by the characters to their respective sides. Consider that Mercutio, who isn't even related to the Montagues, fights as if he is Romeo's brother. Then, only when he is mortally wounded does he realize just how out of hand the hatred has become. His famous line, "A plague on both your houses!" is the alarm that, at last, it has gone too far.
Though a dramatic comparison, it should be noted that entire wars have started as a result of one leader feeling insulted by someone. Gang violence and warfare so prevalent in major US cities could likely be traced back to a series of once petty insults that escalated to full blown hatred. In just two scenes, Shakespeare has captured several common human emotions, desires, and actions that result from what is often passed off as "petty insults."
In Act 1 Scene 1 the Montague's and the Capulet's are fighting and cause a street fight then Prince Escalus warns them that if another street fight occurs the people involved will be sentenced to death. In Act 3 Scene 1 Benvolio and Mercutio are together and Tybalt comes up to them seeking to talk to Romeo. But then Mercutio causes Tybalt to get mad and then Romeo comes and Tybalt tells his to draw but Romeo refuses knowing now that he is married to a Capulet and that they are legally related he now doesnt want to fight. Then Mercution gets mad and challenges Tybalt to fight and they do but Romeo gets in the way and Mercution gets stabbed under Romeo's arms by Tybalt. After Mercutio dies Romeo tries to avenge Mercutio so he kills Tybalt. Then he remembered Prince Escalus laws and runs. When Prince Escalus and the Capulet and Montagues show up Benvolio explains what happens but The Capulet dont believe him because he might be trying to protect the Montague. So then instead of sentencing to death he banishes him from Verona and if he is seen in the city he will be killed. Hope it helps.
Act 3.1 is ismilar to Act 1.1 because in Act 1.1 some Capulets and some Montagues fight, leading to a large fray. Evntually the prince of Verona intervenes, setting down a law for all Montagues and Capulets. In Act 3.1, some Montagues and some Capulets fight and two people killed, the fight braks up and the prince sets down another law, this time on the killers.