What is the specific mechanism that causes death in hanging?

The cause of most hanging deaths is asphyxiation, not broken bones or severed spinal columns. You see those is a hanging with a drop, such as execution by hanging, where very few have that part.

When the person puts their weight on the ligature (rope, wire, cable, sheet, shirt, pants, whatever) it does several things. The airway is blocked, but the carotids are also clamped and stop allowing blood to the brain. This part actually makes you pass out, and then you just can't breathe any more and die.

If the ligature is set too high on the neck, it may not shut off the carotids, which means the subject will then usually fight to keep breathing once they figure out this isn't as easy as they thought. Depending on how they set this up, it may do no good and they die anyway, but sometimes they manage to get loose.

Actually, a severed spinal cord can indeed cause death, if it's sufficiently high up the cord. Above level C2-C3 (the second or third cervical vertebra), you will cut off supply to the nerves that control the heart and lungs, which will be fatal. Below that, those nerves have already branched off so cutting the cord will usually not be fatal, but at high levels it will be.

More importantly, though, in a hanging, the blood vessels to the brain can be compressed, as can the airway itself, and those will be fatal.

Rapid fracture of the hyoid and severing the spinal cord usually results in a very rapid death: seconds to minutes.