The difference is in how the servo motor is controlled via the circuit board (amplifier). The motor of an analog servo receives a signal from the amplifier 30 times a second or at 30Hz. This signal allows the amplifier to update the motor position. Digital servos use a high frequency amplifier that updates the servo motor position 300 times a second or at 300Hz. By updating the motor position more often, the digital servo can deliver full torque from the beginning of movement and increases the holding power of the servo. The quick refresh also allows the digital servo to have a tighter deadband
In a word, nothing. Analog servos can be just as powerful and fast. The only real advantage for most with digitals is they are more consistent. I think the speed of current servos is more a side effect of development than just the fact that they are digital.