Primers aren't the difference (in hardness, at least). Most steel-case ammo has a different type of primer, and sometimes it will be a "hard military style" primer, which is a whole separate issue having to do with hammer force and free-float firing pins ... but I doubt it will damage most centerfire guns' firing pins.
The steel is a very mild steel, but it will not expand and seal in the chamber the way brass does. It is also a bit harder on extractor claws in semiautomatics, although a robust design shouldn't be harmed by the very soft steel alloy used for shell casings.
Take a spent steel case and a spent brass case and put the pliers or vice-grips to them, the difference isn't that much as far as the strength at the extractor's grab point on the rim.
Some guns just don't like it, so test and check for reliability and wear, and if you're paranoid look up how much hassle a new extractor would be (in most cases, pretty easy and cheap).
It is often dirtier than comparable brass ammo, both due to the lack of seal (more crap in places other than down the barrel) and due to using cheaper powders that just make more crud.
And don't let some booger-eating-moron accuse steel-case ammo with steel-core or steel-jacket ammo, which can be harder on a backstop in some cases.