AF447 was caused (partially) by the pilot's misunderstanding that holding the control stick all the way back was gonna make the aircraft go up at the best rate of climb, and not put the control surfaces all the way back.
The AP (autopilot) was off for quite some time when the crash happened. What was supposed to be on though, as always, is the Airbus Fly-by-wire system in normal law, that is in between the pilot's controls and the actual commands sent to the control surfaces (ailerons, elevators...). This system was off (in it's direct control law, more precisely) due to the pitot tubes sending non accurate data, because of the icing problem. The pilot didn't notice that.