How does the two coils having independent wiring? What separates a humbucker from two single-coils going into the same pot?
it is like 2 single coils, except that the single coils are right next to each other and they are wound in opposite ways(clockwise, and counter clockwise), that way the 60 cycle hum from the 2 pickups cancels each other out. the hum canceling has to do with electromagnetic b fields produced by the electro magnets in the passive guitar pickup.

btw you can coil tap yer buckers to make them single coil. another thing people do on strats is wire the middle pick opposite of the bridge and neck so when you switch to the 'between' setting it cancels hum.
Well its because the two coils in thre humbucker are reverse wound, so that makes that the hum produced by one coil gets cancelled by its opposite produced by the other coil
They buck it.

sorry. The others speak the truth. Its like 2 single coils which cancel out each others hum.
WTLTL 2011