From memory; pitot tubes froze over and airspeed became unreliable. Autopilot disengaged and they had a stall warning. One pilot went full nose down, the other wanted full nose up. The latter probably believed the flight envelope protections were still in place which would not let you stall even if you did full nose up. But they weren't because the flight computer switched to alternate law when airspeed was unreliable so they had full control of the airplane and (probably) didn't realize it. Additionally when presented with two different control stick inputs the flight computer just averaged them, and since there was no force feedback there is no tactile sense that you're arguing with the other pilot over what you should be doing. Also only one person is supposed to be in command.
My memory is also that the senior captain was in the third seat and didn't have a full grasp of the situation either.
I did read the air accident report quite a while ago so this might be wrong, but in summary, it's not exactly arguable that the aircraft did the wrong thing (it did as designed minus the freeze issue which is suboptimal but something that can happen) and the pilots didn't exactly do the wrong thing either since they may have little to no experience flying the plane in alternate law (certainly not at high altitude and speed), and it's not unreasonable to be fully aware that they had full control (as opposed to the flight computer mediating and not letting them do something stupid like stall it all the way down to the ocean).
Lastly, when in cruise it's not completely trivial to fly like a Cessna at 1,000ft. At cruise on a big heavy jet you can't go much higher as the air is too thin to fly. You can't go much faster without braking the sound barrier. You can't go much slower as you'll stall. I believe it's called "coffin corner", and while commercial passenger planes do it every day it's not quite as simple as you might think.