How can we relate the society in Verona as Shakespeare presents it in Romeo and Juliet to society today?
The simplest answer to your question is that whether it be as rival families (Romeo and Juliet) or differing in regards to religion, race or politics, we still choose to battle each other -- be it in the city streets or in the battlefields of war.
Human beings have not learned to accept the differences that we continue to exhibit and Romeo and Juliet, by not explaining what exactly it is that has caused this feud between the Capulets and Montagues, does a great job of showing just how inexplicably heartbreaking being at war with one's neighbor can be.
The Prince, the designated ruler of this Venice, stands above the fray and attempts to alternately mediate and solve the violence, but he remains ineffective. It is finally, the dear losses suffered by those "at war" with each other that ends the violence, as is often the case in any human struggle. When we look into the eyes of our enemy and see our own human suffering reflected there, we are finally able to understand that our differences -- be they political, religious or racial -- do not outweigh the common emotional connections we share as human beings.