On the other hand, screws in a humbucking pickup are just screws, but conduct the magnetic field from a magnet placed underneath the pickup. There is another set of magnetically conductive slugs in the other coil, so the combined humbucker reads a larger length of string vibration. This was allegedly to allow him to move the pickups closer to the strings without the problem of excessive magnetic pull on the strings. I've never tried this myself, but it does illustrate how changing the magnetic field of a pickup changes the way a pickup works and sounds. Magnetic material also affects tone.
Ceramic magnets have a slightly harder edge, and are favoured by metal players.
With so many variables it's easy to see why there are many different choices. There are a few established standards, notably the original Fender Stratocaster, Gibson PAF, along with some popular models from retrofit manufacturers like DiMarzio, EMG and Seymour Duncan. Even before you plug your guitar in, its character is determined greatly by the type of guitar woods used, construction methods, string gauges, etc. It is this basic character that the pickup picks up, adding its own colouration. All of the above desribes magnetic pickups, which are by far the most common pickups used on electric guitars.