They were invented over time to overcome the drawbacks of other types. The slot type is the oldest and was in use for many centuries. I believe the phillips came next which improved in the slotted. After that came all the others over the last 100 years. Robertson and torque heads were to improve on the Phillips head and most of the others were originally for proprietary use that were eventually copied.
There are really only a few types of screw head in common usage: flat (slotted), Phillips (star), and in Canada, Robertson (square). There are other shapes, obviously, like Torx and hex heads, but they tend to be more for bolts than screws. There are also some types of screw where the head is specifically designed so that you can’t turn it unless you have a matching esoteric bit, but most people will never see those.
All of them are designed to try to address the issue of being able to apply torque to the screw sufficient to drive it into the material without the head of the screwdriver slipping out (slotted screws) or rounding off and deforming (Phillips screws). Keep in mind that’s rounding off either the screw or the bit. Rounding off a bit sucks because it can happen quickly if you’re a dummy and it’s money in the toilet.
Some context:
Slot: easier to manufacture. The OG of screw types. It sucks, but heck, when making screws was hard, it worked well from a machining standpoint.
Phillips: two purposes. One, the driver won’t just slide out like a slot. It also cams out as previously mentioned which has its advantages. You can still wear the pattern in the screw if you go “nuts”. It is harder to reliably stamp/machine on a screw head than slot.
Robertson: square head. It doesn’t came out (both good and bad). In my experience, it is harder to deform the patter on the screw head, but it can absolutely be done still. Making the hole on the screw head square is also more difficult than slot types.
Torx: I expect it was designed to make removing the screws harder, basically an anti-tampering measure.
Hex head: I am not sure of its origin, but I expect it was also designed for reasons similar to the Robertson.
Other weird screws (triangle, spanner, etc.): anti tampering measures. They’re basically useless nowadays, but some still use them.
EDIT: For the obligatory xkcd, I expect there is some of this too that happened at some point: https://xkcd.com/927/ Someone found something on a design inadequate and figured they’d make a better screw type. Result: one more screw type around.
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