I forgot about bronze. It is not better than iron though, in fact it has very similar properties to the point where if it is better, it's only a very slight difference. It's generally used as a substitute for iron, not to replace it as a superior material.
This is complicated. However, in the real world, bronze was clearly superior to copper, but required rare and/or expensive raw materials, such as tin. If you think of bronze as "copper, plus crazy expensive additives, plus skill, equals superior swords", you're not far from wrong. DF doesn't do a good job of handling relative material abundance, so this doesn't come out in gameplay very often.
Iron was harder to work, usually required more fuel to make, and gave weapons and armor that were inferior to bronze due to problems with brittleness vs. ductility... but iron ore was *everywhere*. Suddenly, equipping a metal-clad, sword-wielding military was within the reach of any reasonably functional despot, probably with resources available in their own little kingdom. Sure, over a long campaign, the iron stuff bent, broke, or shattered a lot; but if you could field several times the number of iron-clad warriors than your neighbor's bronze-clad warriors, chances are you would win before that became much of a problem. In DF, this would only matter when we have a functional armor-damage system.
Steel changed things again... it's a "miracle" technique to turn all that crappy iron low-bidder stuff into a wonder metal, using fairly common other materials. Compared to above, it's "iron, plus cheap additives and extra fuel, plus more skill, equals superior swords". Steel isn't always the best metal to make things out of, but it's usually "good enough", so societies eventually gravitate toward "steel everything", except for special cases.
RPGs and video game RPGs in particular seem to have driven a generation of kids to think that metals have a simple upgrade path, with each improvement being superior to the previous. This was somewhat necessary in the early days of 8-bit console RPGs, but has been an oversimplification for a long time. If "Fluff" is a +2 material, and "Lint" is a +3 material, then a Lint Sword will be better than a Fluff Sword, Lint Armor would be better than Fluff Armor, ... heck, a Lint Cooking Pot would be better than a Fluff one. The world doesn't work that way