Generally; intensity and length of burn.
Of course from brand to brand there are a lot of differences. CCI primers for example burn much better (not sure if hotter, longer, or both) than Winchester or Federal. I had a .433" round ball lodged in the throat of my Marlin 1895 from a Winchester primer failing to set off the powder in a reduced charge load. Never had a misfire with CCI's.
There are of course those that use magnum primers in all cartridges or never use magnum primers and just use normal ones in magnum cartridges. It's a mixed bag.
Personally I use CCI magnum primers (either LR or LP) for reduced loads in any cartridge and all the time for .44 mag and .300WM (largest cartridge I currently load for).