There seems to be an awful lot of generalization that doesn't apply in every case.
Magnum primers:
The main purpose is to make more gas than a standard primer. When you ignite a powder, especially a slow powder, it can extinguish if the starting pressure isn't high enough. In a large magnum case there is more total air space between grains and around them, so the magnum primer is designed to make enough gas to adequately pressurize it. This also makes magnum primers a better choice in some standard cases if you are using a load that doesn't fill the case well, because the magnum primer better pressurizes the extra empty space. I recommended this to a fellow shooting .30-06 Garand loads with about 75-80% case fill at the CMP forum, and his groups were significantly improved.
There are two ways to get a primer to make more gas. You can increase the quantity of your standard primer priming composition or you can make a new mix that has more fuel in it, but no additional sensitizer, which will be a lower brissance mix and can run cooler. There are often now aluminum particles or other material added to throw glowing sparks from the primer to help ignite heavy deterrents.
Alan Jones said that at CCI they had altered their magnum primer mix in 1989 to better ignite the spherical powders then available. (Western Powder says the more modern deterrent coating formulations on the Ramshot line are easier to ignite and don't require a magnum primer.)
Because of the variables in priming mixes and case volumes, Hartmut Broemel points out there are some scenarios in which lower peak pressure results from using magnum primers. For example, when a primer makes enough pressure to cause the bullet to unseat earlier in the powder burn. This causes premature expansion which can actually retard pressure build up. It is usually associated with erratic ignition and muzzle velocities. It is a common problem in the small capacity .22 Hornet that even standard primers are not mild enough and it can sometimes require pistol primers to give best accuracy. The issue also comes up in some pistol rounds even with standard pistol primers. The bottom line seems to be not to try to shove too much primer pressure into small case capacities.
Can magnum primers make higher pressure? They can sometimes do that too. IIRC, in the March-April 2006 issue of Handloader, Charles Petty ran the same 24 grain load of Reloader 10X under a 55 grain V-max bullet in his .223 Cooper Phoenix using different primers. The velocity went from about 3150 fps using Federal 205 (standard) primers to 3300 fps using, I think it was, the Remington 7½ BR match primers. I've forgotten for certain, and can't seem to lay my hands on my copy of the article at the moment. The 205M match primer was also tried and, again IIRC, was next faster after the 205 and produced slightly lower velocity standard deviation, but both were exceptional in this regard. It seems to me the accuracy was no better at 100 yards. The high velocity producing primer had the worst standard deviation, but at 100 yards produced the best accuracy.
(3300/3150)²=1.098 or about 10% higher kinetic energy was in the 3300 fps load. Kinetic energy is proportional to average pressure in the bore (acquired energy=work done on the bullet; force times distance). Since the powder is fast enough to burn completely in the barrel, the muzzle pressure won't be significantly different (a little more due to more primer gas, as little less due to larger pressure gradient behind a faster bullet). So, pretty much that whole 10% increase in the average pressure will be due to what happens at the peak end of the pressure curve only, so the peak pressure was probably up by about 20%. (I went and checked this in QuickLOAD adjusting the RL 10X burn rate to provide both velocities in a 24" tube and got peak pressures of 44759 psi and 53712 psi in a case with 31.5 grains as-fired water capacity, or just about exactly a 20% peak pressure increase. Muzzle pressures were within 2% of one another.) As to accuracy, the higher peak and bullet velocity result in shorter barrel time (1.005 ms vs. 1.086 ms in QuickLOAD), and the barrel time for the faster load was probably just falling on a muzzle deflection phase sweet spot that the slower loads missed. In other words, it tuned the load to something more favorable for that barrel.
I hate working from memory on Petty's stuff and if I can figure out what I did with the article, will post any corrections.
Cup thickness:
This is another variable. The James Calhoun article Ole 5 Hole Group linked to is the only one I recall, and it's from 1995. That's the one showing all large rifle primers with the same cup thickness and small rifle primers varying, and includes the Federal 200 magnum pistol primer having the same cup thickness as a standard small rifle primer from the same maker. But that's an 18 year old article, now. One of the things Alan Jones points out it that primers are changed without notice. An example from closer to that time is when Winchester stopped putting nickel plating on their primers for awhile because of sensitivity complaints. Then the complaints went the other way and they put it back. The thickness and hardness of the two forms would not have been the same. I also have some FC headstamp match cases from back then in .308 whose large primers were not plated. I also expect the Wolf and Tula and S&B and any other primers from the other side of the pond to be some metric thickness that doesn't quite match Calhoun's table, either.
When CCI came out with their military sensitivity spec primers, the #41 and #34 primers, they started with their magnum primers as the base, then devised a shorter anvil with a larger angle between the legs of the tripod to cut down sensitivity. When Federal decided to come out with a non-magnum small rifle match primer with military sensitivity for the AR, I inquired and got an email saying it was done by putting the 205M mix into a thicker cup. They said they did not mess with anvil to make the 205MAR. We know from CCI's experience that simply going to a magnum cup was not enough to hit military spec, so I'm expecting the 205MAR cup to be even thicker. Unfortunately, the shortage has prevented me from getting my hands on any to make a measurement. Nonetheless, we know it's a standard strength primer with a thicker cup than the Calhoun table shows for other standard strength small rifle primers.
Calhoun points out there can be a large range of hardness difference in the brass cups based on how hey work it in manufacturing. However, since increasing hardness decreases ductility, I expect the various makers use a fairly similar hardness compromise to be able to withstand deep firing pin impressions without piercing and also to withstand pressure without piercing.