Luckily, magnets do not permanently damage analog watches. As soon as you move the watch away from the magnet, the stepper motor should start working normally.
Other types of watches may suffer from magnetism. For example, mechanical watches have small hairspring coils. When exposed to magnetism, the hairspring coils stick together.
When the coils stick together, the length of the hairspring is effectively shorter. This causes the watch to gain time. You may notice that each day your mechanical watch is slightly further ahead compared to the previous day.
In some cases, the hairspring coils in the mechanical watch may become completely stuck due to magnetism. This keeps the watch from working at all.
Part of the problem comes from the various steel components in the mechanical watch. Steel can retain magnetism after leaving a magnetic field. As mechanical watches can retain magnetism, they may need to be demagnetized occasionally.
Watch manufacturers have tried to combat magnetism by replacing steel components with components made from other alloys.
For example, instead of steel, a watch manufacturer may use aluminum components to limit the watch’s ability to retain magnetism. These watches are often marketed as “magnetic resistant” watches.
Digital watches do not suffer from the same problems, as they do not have motors or mechanical parts. The electronic components and digital display are not typically affected by magnetism.