What specific event in Shakespeare's Romeo and Juliet displays how not letting go of old traditions affects the younger generation negatively?
The best event that displays how the traditions of the older generations are negatively affecting the younger generation is actually the brawl that takes place in the very first scene.
What is interesting about this brawl is that the Capulet servants actually have no motive for causing it, other than that they are aware that their master is engaged in a feud with Montague. They are walking down the streets of Verona at a peaceful period of history carrying swords and "bucklers," meaning shields, on some errand, which alone is unusual considering that they are just servants. Due to their swords and shields, plus the feud, they feel inspired to engage in a conversation that challenges each other's bravery and manhood. We see Sampson defend his bravery when he says, "Gregory, on my word, we'll not carry coals," meaning not allow ourselves to be humiliated by not drawing their swords (I.i.1). Sampson refers to their swords in his next line when he says, "I mean, an we be in choler, we'll draw" (I.i.3). The term "choler" is another word for "anger" (Random House Dictionary). In other words, Sampson is arguing that if they find themselves angered, probably by the Montagues, they will draw their swords and challenge them.
Gregory later points out that because the feud is between Lords Capulet and Montague, the feud is also between their men, meaning their servants. We see this in his line, "The quarrel is between our masters and us their men" (I.i.17).
Hence we see through this passage that even though they have no cause, the families' servants are willing to get into public brawls simply because their masters are feuding. The older generation's tradition of the long-standing feud is provoking the younger generation to unnecessary wrath and violence, which is reeking havoc on the younger generation.