The old paper shells had two wads, the over powder wad, which was made of a sort of felt or fiber that cushioned the shot and provided a seal to prevent powder gas from getting into the shot column in the barrel and scattering the shot as it exited. The other wad, the over-shot wad, was made of a stiff cardboard that was supposed to break up on firing and thus also not get in the way of the shot. In practice, the over-shot wad did sometimes interfere with the shot column, creating a "hole" in the pattern that supposedly let game get away. (Of course, it also provided an excuse for hunters who missed, but that is another story.)
The sides of the shell were crimped over to retain the wads and shot and the top part of the over-shot wad was printed with the dram equivalent and shot size of the shell.
Another problem was that the shot contacted the sides of the barrel on the way out, and some shot would be deformed, causing it to fall away from the column. One of the first uses of plastic in shot shells was a shot wrapper, simply a piece of plastic rolled into a tube and inserted between the two wads; the effect was of a "cup" which cushioned the shot as well as protecting it from the barrel. An added benefit was that the shot wrapper protected the barrel from the shot, reducing leading.
Finally, the shotshell companies developed the plastic shot cup, which combined the over-powder wad and the wrapper in one unit, and about the same time, the crimped seal, which did away with the over-shot wad. At first, the inside shell base was still a separate part, but then it became part of the shell. Going to plastic made things a lot better for hunters and shooters in most ways, since the new shells were more accurate, shot better, and didn't deteriorate in wet weather like the paper shells did.
Naturally, there was a down side. The old paper shells were degradable and soon disappeared in the dirt and mud of hunting areas. The plastic shells do not, and if not picked up will litter the area forever.
Another, lesser, problem, was that the old paper shells were waxed. When fired, they left a wax deposit in the chamber, and hunters could skip rigorous cleaning in that area. The plastic shells left no wax, and chambers left without cleaning and oiling began to rust. Many hunters blamed some unknown "deposit" from the plastic for rusting chambers; in fact it was the lack of a wax deposit from the paper shells that was to blame.
So, plastic certainly gave us better shotshells. But the shotshell companies weren't totally altruistic. Remington-Peters, Winchester-Western and Federal were all divisions of chemical companies that made (you guessed it) plastic. What better product is there for continued company prosperity than one that is literally blown away.
Jim