In essence:
- Coil Tap shorts out one coil, turning a humbucker into a single-coil pickup, complete with S-C noise problems.
- Series connection has the coils connected one-after-the-other. This results in relatively high output, and humbucker noise cancellation.
- Parallel connection ties the two 'low' ends together and the two 'high' ends together. The output voltage is lower, and the sound is a bit thinner and more clear. Something like a single-coil sound, but retaining the noise-cancelling humbucker benefit.
Any of my guitars that I've installed aftermarket humbuckers in have been set up with Series/Parallel and Phase switching. That gives plenty of options on your sonic palette. Going overboard with switching permutations can often produce a confusing array of combinations, only a few of which actually sound good.