It was because Pete attracted everybody`s attention and was the most popular of the 4 guys. Bill Harry mentions this fact in one of his interviews, he says that definitely and undoubtedly Pete was the most popular of the 4.
After one performance attended by Paul`s father the guys left the stage and were surrounded by the fans, mostly girls. However, the three were quickly set free and could go home, but Pete was beset by a much larger number of fans and had to stay with them long. Paul`s father, who saw it all, didn`t like it, and later approached Pete to reproach him about it and to tell him that it was unfair. The reason he was against Pete was that he was jealous.
This is one thing written in books and I am sure I`ve read about it more.
Maybe, now it is my suggestion, Pete and Paul were on one wave, singing mostly songs about love at that time, and were kind of rivals? I will look up the facts in the books.