It could be argued they mean the same thing, but typically the “inspection” part is more formal, in that you are checked to make sure you have certain equipment that is needed. Some equipment is common to everyone– canteens, packs, ammo pouches, and so on. Some equipment is specialized– a machine-gunner and his assistant must have extra ammo specific to the machine gun, for example. Or perhaps certain people are tasked specifically to carry radios, a GPS, a map, or so forth.
A pre-combat check would be to make sure that these items are not only present, but in proper functioning and useful order. For example, a canteen is useless without water, ammo pouches should ideally have magazines full of ammo, radios and night-vision goggles should have batteries and some spares, etc.
PCCs and PCIs are important, and Joe Rock has been handed a new task: make sure his buddy, Specialist Ricardo, quits filling his Camel-Bak full of coffee. Not only is it bad for the Camel-Bak (and Ricardo’s kidneys), coffee doesn’t really hydrate you very well and, of course, the guy with the machine gun is the last guy you want all sketchy on caffeine. PCIs and PCCs are typically done by the junior Sergeants in a squad to make sure everyone’s fully ready before heading out to a mission. Once PCCs and PCIs are done, you’re out the door and it’s too late to turn back for an item you forgot.