It's like anything else, each manufacturer produces a different product than the others, and even within a given brand there are variations between lots. There are chemical differences between the explosive mixes, the way they crystallize as they dry, the manner in which wet mix is placed in the cups (done by hand by the way). Then there are differences in the cups, thickness, hardness, dimensions, etc. At one time Remington's DuPont manufactured primers came with two legged anvils, while everyone else used three legged anvils (see below).
At the time I found the Remington product to deliver a slight accuracy advantage across the entire primer line. Now Remington brand carries three legged anvils just like everybody else, not that some other difference wasn't the cause of their better performance. Today the Remington 7-1/2BR primer is my go to choice for the .204 Ruger and .223 Remington for most powders, with the Federal 205M the choice for those cartridges when using Reloader 10x powder. I find that Wolf standard large rifle primers work really well in the .243 WSSM with Ramshot Hunter, producing very consistent results and amazing groups, but they require more force to seat in cases. Winchester standard large rifle primers, still marked as being made by Olin, are another choice in my inventory for the WSSM, just as accurate as Wolf.
Then there is the question of who is manufacturing the brand of primer selected? Like many things, tires and automotive batteries come to mind, primers can be contracted out and rebranded. CCI makes a lot of rimfire ammo sold under other marques for example, and Norma cartridge brass is rebranded and sold as another premium brand. Loads from 20 years ago may have been developed with brand X primers that are not the same as brand X today.
So the answer is that the primer used in the recipes is important as a starting point in load development. The dangers that exist are found in swapping out a magnum primer for a standard primer in certain published loads of ball powder, especially in handgun magnum loads, or conversely, substituting a magnum primer for a standard primer in a near maximum load, especially in overbore capacity bottle necked cartridges. Ball powders have deterrent coatings applied to control burning speed, and generally need a more brisant primer for consistent ignition. I learned this with hangfires and excessive pressure in the .45 Colt loaded with W-296 when the only change was a standard primer for a magnum primer. In substituting a magnum primer for loads calling for standard primers the result can be excessive pressure, but in my experience it usually degrades accuracy with more open groups, excessive pressure or not.
If a recipe calls for a standard primer then stick with those, and vice versa. Try the load exactly as published, if it works satisfactorily, you are home free.
Generally, one experiments with different brands of primers to develop the best accuracy with any given powder, bullet and cartridge combination. In some cartridges and rifles the mechanical properties of the primer may be of the greatest importance, an example would be the use of CCI No.34 and No.41 Military Rifle Primers designed with thicker cups to prevent slam firing in semiautomatic rifles. Handloading is all about making the most dependable and accurate ammunition at the lowest cost. It is also about experimenting, tinkering and obsessing over minutia, that's why its more than a hobby.