In general, for any primer type, magnum primers make a larger quantity of gas to pressurize the case adequately to achieve consistent powder ignition timing. As a result, they are used in larger capacity cases or in medium capacity cases when the load density leaves a lot of unfilled space in the case and so needs more gas to reach the burn starting pressure. In rifle, it is not uncommon for magnum primers to produce the best accuracy and lowest velocity variation with charges that don't fill the case well, but for a standard primer to do better when a charge of the same powder fills the case enough. Some powders, particularly those with heavy deterrent coatings, can require extra starting pressure.
In addition, most domestic primers these days throw a shower of white-hot burning metal particles to help with starting through heavy deterrent coatings. The addition of metal dust appears to have begun in earnest with CCI magnum primers in 1989, when they reformulated them for the older St. Marks spherical powder formulations. Today the same spark shower is produced by many standard primers. As a result, in some instances, especially in shorter pistol cases, the standard primers ignite those slower, high deterrent powders just fine. Some of the primers from Europe and the former Eastern Bloc countries do not make the spark shower. The bottom line is you have to test with the primer you want to use to see what it does.