Clothes Dryer electrical 240v between hot and netural, 0 between other hot and neutral
It sounds like the neutral connection is bad, but not totally open-circuit.
As I understand it american driers normally have the high-power stuff connected between the two hots and the control system connected between one hot and the Neutral.
With the dryer disconnected, the bad neutral is enough to give you correct voltage readings on your high-impedance multimeter, but when you connect the driers controls. it drags the neutral up to the same voltage as one of the two hots.