While the reason for the feud is never specifically mentioned, we as readers may infer that is stretches back many generations, as the author tells us that the feud is both old and bitter, stemming from past wrongs of other eras. Much like the Hatfield-McCoy feud seen in Southern- American History, this feud is one of names: Capulets don't like Montagues because of their identity, and Montagues reciprocate that emotion. While neither side really elaborates upon why the hatred is still present, they continue to engage each other in acts of aggression and conflict, much like the "biting of the thumb" and physical altercations that occur throughout the play.
The reason for the conflict between the Montagues and the Capulets is never revealed in the play. The play's Prologue reveals that there is an "ancient grudge" between the two families. Additionally, Prince Escalus mentions their conflict and its effects on Verona, but again there is no reason given for it. This conflict is brought up several times throughout the course of the play but the true purpose for their disagreement is never revealed.
As mentioned in previous answers the reason for the feud is never specifically mentioned, however we are told the futility of the conflict as suggested by an ancient grudge. The word ancient suggests that the grudge started long ago, meaning the real reason for it is long forgotten. Therefore the grudge is petty though the resulting conflict is not.