Slotted head:
The pro is that they match when restoring old antique furniture. The cons are that it is hard to keep the screwdriver tip centered, especially with an electric screwdriver, so it is easy to gouge out the wood around it.
Phillips head:
The pros are that they are self-centering, ubiquitous, cheap, you can get bits everywhere for them, and there are only a few sizes to worry about. The big con is that the head strips out easily. They were actually designed to prevent over-tightening by allowing the screwdriver tip to spin when they got too tight. Of course, with electric screwdrivers/drills, that just means we spin the bit and strip out the head.
Square/Robertson head:
The pros are that they hold the tip/bit tightly, the head is just about impossible to strip out, and there are few sizes to deal with. The cons are that they're expensive (at least in the U.S.) and can be difficult to find.
Torx/Star head:
These are the ones with the six-pointed star-shaped hole. They are even harder to find than square head, so they don't come up as an option for many woodworkers.
Allen head:
These have the hexagon-shaped holes. They are typically used in machine screws, not wood screws. The pros are that they hold the bit/tip tightly, are very difficult to strip out, and the drivers are easy to find. The cons are that Allen head wood screws are uncommon, and there are lots of different size Allen wrenches, which can make them a real pain to work with.
