To clarify a bit:
The traditional style goes back a long time. Unfortunately, I'm not sure how long, but sometime in the early 20th century. This includes the "duplex with a screw in the middle" for receptacles and a small slot for a simple toggle switch for lighting (can be used for appliances too).
Sometime in the 1970s, Leviton started selling Decora switches and receptacles. Unfortunately, I am having trouble finding an exact date. The Leviton history page has it lumped into the "1970s - 1980s" section. I first recall seeing a Decora light switch in 1980.
The neat thing about Decora is it provides a fairly large "do anything with it" rectangle. For a light switch, that means a nice big surface to press (unless split into 2 or 3 switches). For receptacles, that means no screw in the middle. But I think the most common reason to switch to Decora (or to any equivalent size from any other manufacturer) is GFCI. A typical GFCI duplex receptacle has the TEST/RESET buttons right where the screw and middle piece of the cover would be. It just doesn't work with the traditional style. And once you change one receptacle for GFCI, it is only natural to start changing others to use the same style, and in the end having one style cover plate is actually a good thing.
Electrically, assuming there is no GFCI involved, there is no difference between the traditional duplex receptacle and Decora-style duplex receptacle, or between the traditional "flip switch" and a Decora-style switch.