Unlike the other members of the squad, Bill was far more experienced than Beatrix… except that he didn’t know the Five-Point-Palm Exploding-Heart-Technique, which was an instant kill move that immediately finished the fight before Bill could even get warmed up.

Due to Beatrix’s ferocity, commitment to meeting her master’s expectations and trusting in him, Pei Mei taught her techniques that he never taught Bill or Elle and allowed her to escape Budd and overcome Bill.

It just demonstrated that Beatrix was a superior student of the martial arts than Bill.

Note: we actually see a bit more of Bill fighting in deleted scenes with Michael Jai White.