I'd say the main benefits of metal arms and armor are that they don't break as quickly, and that they can be strong and light at once, rather than one or the other. Stone, bone, tooth, and obsidian weapons can be as sharp as or sharper than metal blades, but they take a lot of work to make and break very quickly. Wood, fabric, and hide armor work well--Aleksandr Baranov said Tlingit hide and wood armor could stop his men's musket fire!--but can weigh more than metal for a given level of protection, and get damaged quickly.
Swords don't really deal with armor except in some specific cases by stabbing, since even thick clothes can stop a cut. Instead you just target the places where there's no armor. The conquistadors often swapped their mail for Aztec and other native cotton armor, giving a variety of weird reasons I'd never heard used before, like "the arrows could splinter and get through our mail" or "arrows would bounce off and hit our comrades next to us but they safely stick in cotton armor." Maybe they found cotton armor easier to repair or replace in an area that made it but which made no mail. In any case, they liked it, and many conquistadors used leather adargas for shields rather than metal rodelas. Steel swords are nice because they can cut, thrust, and give decent reach while being light and durable, but wooden swords, macuahuitls, and stone maces are still useful--and in any case most of the killing in battle was done by cavalry lancers and, probably, by native allies. It's also interesting to note that Coronado's expedition muster roll listed many conquistadors armed with "arms of the country," probably meaning "native Mexican weapons," which would not have been metal but wood and obsidian.
In days of yore, I had a thread on armor in the mostly-dead World History subforum, and it had a post on Tlingit wooden and hide armor. I'll dig it up later.