Any attempt to discuss weapons without discussing armor is incomplete, because the two go hand-in-hand. The change in weapons over time isn’t just driven by changes in the technology used to make them and the social contexts in which they get used; it’s also driven by the changes in the defenses mustered against them. As armor gets good at defending against one kind of attack, people switch to attacks that will exploit some other weakness. (A point made very nicely by the movie The Dark Knight, when Batman gets chewed on by dogs.) And as attacks improve, armor will be developed to defend against those new threats.
All armor has to make choices — or rather, the people who make and wear it do. Kevlar vests can protect against bullets, but they’re not as effective against cutting attacks (though still better than ordinary clothing!). Chain mail deflects slashing fairly well, but piercing weapons can sometimes break the links open, and it doesn’t do a lot to help you against blunt-force trauma. Rigid protection like plate armor is good against practically every kind of attack, but it sacrifices mobility in some obvious ways: great for things like helmets, not so much for the parts of your body that need to bend. For those, you’ll need specially shaped pieces with the gaps filled by some more flexible material . . . and those gaps are going to be more vulnerable.
Weight is a non-trivial consideration, especially in the days before modern materials engineering let us develop strong, lightweight alloys and plastics. A suit of late medieval armor, with all the specialized pieces to protect different parts of the body, could weigh anywhere from thirty-five to fifty-five pounds. Now, this wasn’t the cumbersome, half-immobilized tin man of popular imagination; well-made armor allowed its wearer almost complete freedom of movement. But it’s one thing to be capable of doing a dive roll in your armor, and another to haul that extra weight through an entire battle: combat is incredibly tiring, even in short spurts, and weighing yourself down with a second skin made of steel just magnifies that.
Mind you, not all armor is made of steel. But leather armor isn’t really what games would like you to believe: while leather has been used in many parts of armor-making, as a backing for rigid pieces or a connector between different parts of the armor, on its own, it isn’t actually much use as protection. It works reasonably well against incidental dangers — that’s why motorcyclists often wear it — but even in hardened form, a deliberate attack can too easily cut right through it. Still better than nothing, of course, and the same is true of wooden armor, or the linothorax of ancient Greece. But on the whole, metal was historically your best bet for protection, whether it was bronze, iron, or steel.
But note the “well-made” qualifier a little while ago. Unlike a weapon, which can usually be handed off to another person without difficulty (six-fingered swordsmen notwithstanding), armor has to be fitted to the wearer, or it might hinder as much as it helps. Something like a chainmail shirt is reasonably forgiving in this regard, and to a limited extent you might be able to strap things on in a way that mitigates bad fit, but specialized plates that are too long or too short, too narrow or too wide, can only be altered with the assistance of a skilled blacksmith, or replaced entirely. So while your characters might loot the bodies of the dead for gear, they’re liable to wind up like barefoot John McClane in Die Hard: frustrated because what they’ve taken turns out not to fit.
Video games also lead one to believe that people regularly went around in armor all the time. In an active combat zone, that might be true, but bear in mind that armor is not clothing: it’s some combination of heavy, sweaty, bulky, noisy, chafing, or just plain uncomfortable, and it requires a fair bit of maintenance. (Which, come to think of it, is also true of some clothing . . .) It also suffers from the same social pressures as weapons, which is to say, wearing it in the wrong context will invite blowback: are you implying that you don’t trust the people around you?
As a result, people wear what they think they will need. Early police in London wore two pieces of protective gear: a rigid helmet to guard themselves against being coshed over the head, and a rigid collar to defend against garrotes. The technology available to them wasn’t bulletproof, and the calculation of armor’s detriments vs. the risk of being knifed came down on the side of not being armored. Modern sports gear protects the head if there’s a significant risk of traumatic brain injury (football, cycling), but not if that’s unlikely (tennis, swimming). Men may wear athletic cups to protect their groins, but women generally don’t, because a blow to that area is not nearly as painful if your reproductive organs aren’t external.
Credit where credit is due, though: for all that I’ve ragged on games for ignoring the realistic restrictions on weapons and armor, most editions of Dungeons and Dragons have included rules describing how long it takes to put on different types of armor, which is relevant if your characters find themselves needing to put it on in a hurry. And they also specify that sleeping in armor is not properly restful, to avoid players claiming their character just never takes theirs off.
Unless the armor is magic, of course. Speculative fiction opens up the door to say that some or all of these considerations could be solved, either through magic or through advanced technology. Your armor can weigh nothing and be invisible and allow you perfect mobility while also covering you from head to foot with no gaps (and not asphyxiating you in the process), because it’s made of some super-special material or consists of a forcefield.
But in contrast to my usual refrain of “we can do so many narratively interesting things with magic or technology!” . . . I’m not in favor of letting the speculative element act as a “get out of jail free” card on this. Perfect armor isn’t interesting, because it means there’s no risk or tradeoff the character has to accept. In one of the Pathfinder games I’m playing in, I could have paid for an enchantment on my character’s armor that let her get a full night’s sleep in just two hours — but that would have meant no more scenes where she nodded off on the shoulder of her vigilante ally/love interest because she was trying to live three lives at once. I liked those scenes, so I spent my character’s money on other things.
That isn’t to say you can’t invent some speculative twist to improve on your character’s armor. Just that it should still carry some kind of restriction, whether that’s a real-world one like weight or mobility, or a made-up one like directly drawing on the energy of the wearer for its effect. And remember that it should interface with the weapons they’re likely to encounter! The armor your characters wear should seek to defend against the most probable kinds of attack, while the weapons should seek to exploit the most probable vulnerabilities. It’s the original arms race, and it makes for good story.