The differences as I have been told...
I would't count on all primers of a certain grade to be identical.... that is, say, all large pistol standard primers. Each company may have their own recipe for the mixture, one companies mix might be a bit hotter or not as hot, that is why you shouldn't change brands of primer (especially at or near max loads) without due caution (back off the powder & test up again) It may be fine, it may KABOOM your gun!
Say large pistol, for example..... naturally they are spec size, so all will fit all cases the same, all will fire the round..... the main difference is the mixture, the quality of the mixture, the potency of the mixture, the consistency of the mixture & the skill of the dude applying the mixture.
Evidently, the high quality bench rest & match primers, the person applying the mixture has to have 5 years experience at doing the job to become qualified to work on the top end primers (they have to really know what they are doing)
I reload 44 mag, normally I use large pistol mag primers, but I havent seen any lately, nor even large pistol primers, so I am forced to use large rifle primers..... I really cant tell the difference, they work.
Primers are designed with a mixture to 'best/efficiently' ignite a certain amount of powder expected to be in a case. For example, large pistol primers are designed for say, 44 special/Russian.... mag for the magnums (more powder) so bigger flash. So, if you use a LPmag primer in a 44 russian, and you are at the max load with the large pistol primer, the mag primer may/will take the load over the max load reccommended. If you are at the max load with 44 russian/special with a mag primer, and you start using (all you can get) large pistol primers, the load is/will become under the max load. (you might need to make up the difference in powder)
If you use a large rifle magnum primer in a 44 special..... you will have to back off the powder a fair bit, as the LR Mag primer is designed to flare up big enough to light up a huge rifle load of powder (probably a 44 special with a LRMP would fire a shot without powder?)
From what I have been forced to use, large rifle primers (not mag) appear to be about the same as a large pistol mag primer (they light approximately the same amount of powder)
As someone said earlier, and as I said above.... the worry is if you are at or near the max load with standard primers, then have to use mag primers (all you can get or by choice) the mag primer will boost the pressure conciderably, so be aware.
When the difference is most noteable is when extreme accuracy is required, for CAS and plinking, any primer that makes my gun go bang & hit a plate is fine by me.... I just use a full case of Trail Boss anyway.... no probs.
If you are shooting targets at 500 yards with large rifle, or hunting at longish distances, choice of primer and matching powder equality of everything to the absolute becomes critical. If you have to use a less than desireable primer, just make sure you dont over do the load, and keep the rounds all together marked in batches, so at least they are constant.
Oh, and one other difference between rifle & pistol primers can be the shape of the anvil & the hardness of the primers metal cup.... generally rifles have a heavier hammer/pin strike than pistols (not always) so the pistol range of primers are designed to go off with a lighter strike from the hammer (the metal cup is softer & anvil/mixture to suit). If your pistol has a weakish hammer strike, it might have trouble setting off a rifle primer (might need a second strike)