What made Herod the Tetrarch so angry with John?

After the death of King Herod the Great, his kingdom was divided amongst his sons. Herod Antipas became the tetrarch of Galilee. Antipas imprisoned John the Baptist because he disapproved of Antipas' marriage to his own brother's wife, who divorced her husband in order to marry Herod Antipas. According to the Gospels, Antipas reluctantly had John the Baptist beheaded at the capricious request of his daughter, during the time of Jesus in Galilee.

However, we have one other source of information on John the Baptist. The Jewish historian Josephus said that Herod Antipas had the Baptist killed in quite different circumstances, because he feared a rebellion. The relevant text from Book 18 of Antiquities of the Jews:

"Now many people came in crowds to him, for they were greatly moved by his words. Herod, who feared that the great influence John had over the masses might put them into his power and enable him to raise a rebellion (for they seemed ready to do anything he should advise), thought it best to put him to death. In this way, he might prevent any mischief John might cause, and not bring himself into difficulties by sparing a man who might make him repent of it when it would be too late. Accordingly John was sent as a prisoner, out of Herod's suspicious temper, to Macherus, the castle I already mentioned, and was put to death."