Well, in the U.K. we call them 'hexagonal' and 'bi-hexagonal'. I know these are big words, but they tell you a lot more about the socket!
Hexagonal have one hexagon, and need to be turned through 60 degrees to re position.
Bi-hexagon have two hexagons superimposed, so you only need to turn the socket through 30 degrees to re engage. Note that they are designed to turn hexagonal fasteners. They had been around for decades before anyone thought of a 12 point fastener!
So what's the point of a bi-hex? Well, when socket spanners / wrenches were first introduced, which was actually in the 19th century (so sorry Snap On, you didn't invent them) ratchets were expensive, and not always very durable, so a lot of turning was done with a Sliding T, Breaker bar, or simply an 'L' handle. With these handles a bi-hex socket makes a lot more sense.
Up till the late 80's, I don't think I'd ever seen a 'modern' hex socket. The only ones around were ancient things. My Dad certainly didn't own any, or my Grandfather. Bi-hex worked fine.
What changed, was the introduction of crappy fasteners using soft metal. In the U.K. I think Ford were the first to start using these. I remember trying to switch a radiator on a Mondeo in the early 90's. The bolts were well crusty and also the new soft ones. I rounded two before admitting defeat and buying some hex sockets.
So, it all depends on what you work on. Modern stuff with ****** fasteners, you're going to need hex. Older stuff, or stuff with decent fasteners, bi hex will do fine. Of course, if you find 12 point fasteners then bi-hex sockets are a must. Personally, I only own hex sockets in metric sizes. If I work on something in AF sizes (to use British terminology, SAE to most here) it's usually a classic British vehicle or an aerospace component and the fasteners are decent. Ditto if I need Whitworth, mostly machine tools with even harder fasteners!
Not that 'bi-hex' is different from 'Tri-square' (the angles of the corners are different) despite both having 12 points) so no, a bi-hex is no good for square nuts. Snap On sell some very nice 'bi-square' sockets for that purpose.