I'm not an expert on metallurgy, but I do know something about the use of these materials and the societies that use them. I'm pretty good with Greco-Roman, so I can offer some views from their world.
First, it is true that bronze is harder than most irons. Unfortunately, I think that statement is taken out of context. Hardness in absolute terms may not be that important; metals have a lot of important properties, including their strength to weight ratio, their abiltiy to avoid shattering, (brittleness), their abiltiy to be cast and formed (ductility), resistance to rust, nad potentially other properties. Although strong, bronze is heavy, very heavy, and shatters, so its hardness is only part of its overall use.
Indeed, bronze is so heavy that it is most useful in formation approaching hoplite warfare, where the strength was in formation. Greek soldiers running away would throw off their panoply, breaking formation. About the worst thing you could possibly call a hoplite warrior was a 'shield thrower', as they would shed their shields as they ran away. The famous saying was that Spartan women would tell their sons. 'Return with your shield, or on it!'. Meaning you die rather than throw your shield! Hoplites caught out of formation were vulnerable because of their lack of mobility.
For bronze, weight also was a problem in its attachments. Real life weapons (not fantasy settings) were usually 1 -3 pounds during the Peloponnesian War. Having a heavy 'weapon' attached to a wooden platform could be problem. Bronze swords tended to be poor weapons, and hoplite armies were dominated by spearmen.
Brass does not make good weapons. Brass is terrific in terms of rust resistance, but it isn't strong enough for weaponcraft.
As stated by others, iron and steel and forms can be complex. Bronze came from tin and copper. Iron was complicated. Iron (like copper) is found naturally as an oxide. If the oxide is burned off, the burned off material produces slag. Taking the oxide from the metal is indeed the process of smelting. It takes about 1000 degrees centigrade to melt copper; tins melting point is very low. Even the lowest melting point irons require 1300 degrees or so (I may be a little off here.) Since oxidation occurs at higher temperatures, the key to smelting good iron weapons, making them steel like, was to apply the heat with minmal oxidation. Obviously, this isn't easy and is the key to making the process work.
Smelting used charcoal, and the Romans deforested a substantial amount of the forests in the ancient Mediterranean producing it. Mixing some slag in small percentages could be good for industrial applications as it makes the metal more rust resistant.
The smith then hammered the resulting metal into shape. If all of the oxide is removed, the resulting metal is called pig iron. Pig iron has a high carbon content.
The carbon gives steel its hardness. Generally, pig iron in pure form is harder than steel. (Again, don't put too much emphasis on hardness!) The steel we talk about today has carbon that is up to about 2% by weight, which has a wonderful combination of tensile strength, rust resistance, ductility, and hardness. The more carbon, the harder and the lower the melting point but again the more britlle. Higher carbon irons are sometimes called 'cast iron' since they are easier to melt and therefore cast into shape.
At the molecular level, the carbon makes the iron harder by keeping iron atoms from sliding through to the crystal structure. In industrial applications, the amount of carbon added depends on which properties are most important.
The Romans produced good iron weapons and armor. They obviously had some good smiths. Armor, in particular, had to be light, hard, and not brittle. Apparently, the Romans after the pig iron was created banged the carbon out on a small flame with low oxygen.
Roman armies were far more flexible than Greek armies intheir phalanx formations. They had iron weapons and armor and worked in flexible formations. However, a Greek phalanx with bronze spears and armor moving forward had greater straight on power.
My view is that iron was the superior weapon for warmaking if the smiths had the ability and energy to melt it and cast it well.
As I said though, I'm not a matallurgist and I may be guilty of assuming their success. But iron is a lot harder to work, and I doubt they (the Romans) would have used it if it weren't better, especially after they conquered enough of the world to use either.
Best wishes,
Breunor