I've heard many people say that a digital autopilot is better than its analog equivalent. How do the two differ in design and the way they control the plane?
Early autopilots would only control one degree of freedom. Starting with the yaw damper of the Me-262, they were a simple feedback loop involving a gyro, a single control surface and some damping circuitry. Example: A roll gyro was coupled to the ailerons and kept the wings level. Commanding a bank angle would simply shift the zero point of the gyro to a different angle, and some damping made sure that the movement was slow and did not overshoot.
This, however, would neglect any cross coupling. Banking flight needs more thrust to keep altitude, but the roll autopilot would not "know" of engines or throttles, just ailerons. The same was true for analogue pitch and altitude hold systems which had a simple feedback loop with the throttle and the elevator, but no other controls.
Modern digital autopilots, starting with the one in the X-31, control all axes in parallel and use a multitude of sensors and feedback loops to control the whole aircraft in one go. This is much easier to achieve with digital control, and gives smoother responses since cross-coupling effects can be anticipated and corrected before they manifest themselves.