What is the Prince’s importance in Romeo and Juliet?

Prince Escalus in Romeo and Juliet is the voice of law, authority, and reason. His role is important in the play because we need to have someone who stands above all the endless turmoil, conflict, and passion engendered by the feud between the Montagues and the Capulets. The character of the Prince alerts us to just how dangerous and out of control this feud has become, how much instability and disruption it has caused to the city of Verona.
It says something about the nature of the feud that even a powerful authority figure cannot stop it, despite his best efforts. There's something almost primal about this conflict, something that taps into the innermost depths of the soul. The Prince is the law in Verona, but even he cannot control custom and its manifestations, and feuding is a social custom that goes back centuries, as old as law itself, if not older. It has a life of its own, and no amount of interventions by the sovereign power is going to make any difference; in the face of tradition, the law is practically impotent. In Verona, honor and blood are more important than formal legal power. It will take a tragedy to bring this feud to a close, and so it does.