Hence, if you're having an argument about which would have been more useful in real life, then the answer is that bronze is superior to crude iron, but inferior to well-made steel. It's all a matter of what quality of metal you can actually produce.
The first practical metal tools and weapons were copper (Cu). They are fairly functional as axes; an archaeological test of a replica of
Ötzi the Iceman's
copper-bladed, yew-hafted axe showed it could fell a yew tree in 35 minutes without needing to be resharpened. Note that this dates to about 3,300 BCE; and the copper had been smelted from ore, cast, and then cold-forged (hammer-worked) to get the final edge. Copper daggers also existed in various societies. The two problems are that copper is rare, and it doesn't have the properties needed to make long blades.
Various alloys of copper were eventually developed (in some cases perhaps naturally or accidentally), that had superior properties. Before the modern era, almost all of these with better tool/weapon performance depended on significant amounts of either tin (Sn) and are generally described as bronzes; or zinc (Zn) and are generally described as brasses. Bronze is significantly better than copper for tools and weapons; the problem is that it depends on tin, an even harder to get element than copper.
A quick comparison of elemental abundance in the Earth's crust, from various references:
Iron (Fe): 4.10% to 6.30%
Zinc (Zn): 0.00700% to 0.00790% (from 519x to 900x less common than iron)
Copper(Cu): 0.00500% to 0.01000% (from 410x to 1260x less common than iron)
Tin(Sn): 0.00022% to 0.00023% (from 17826x to 28636x less common than iron)
So, if you adjusted DF's material odds to be representative of average crustal abundance, and then so that you got useful amounts of iron in *every embark*, you'd get useful amounts of copper or zinc in about one of every 700 embarks; and useful amounts of tin in about one of every 23,000 embarks. The majority of DF players would have never personally generated an embark with copper; and tin-containing embarks, especially those also with copper, would be the stuff of epic threads and legend.
The value and rarity of tin is what drove a significant fraction of early trade and exploration; and in many cases has shaped modern society at a deep level. As just one example, the spread of the Phoenician "abjad" which is the ancestor of all major phonetic alphabets was driven partly by their monopoly on the secret of trade with the tin-bearing regions of Cornwall in Britain, a long and treacherous voyage in rowed galleys from their Canaanite homelands on the Eastern shores of the Mediterranean.
Advanced bronze smiths learned to vary the alloy and hardening content of different parts of a sword to get the best results; the
Sword of Goujian is a fascinating example from around 450 BCE. The majority of the weapon is 80% copper, 19% tin; but the cutting edge is 57% copper, 30% tin, 9% lead, 3% iron; and the reinforcing central ridge is 42% copper, 43% tin, 6% lead, 4% iron, 6% sulfur. This gave the main blade the resilience and shock absorption of a high-copper alloy, with a much harder and more easily sharpened cutting edge and a much stiffer supporting ridge. Performance would have been significantly better than a monolithic sword; when you add in the elaborate patterns in multiple alloy colors and engraved writing, this is clearly a ☼Bronze Sword☼ at the least.
Discovery of the techniques needed to smelt even crude iron to produce marginal weapons was an enormously disruptive game-changer. Suddenly, instead of having to row over 4,000 miles to make a sword, you just had to find some reddish rocks in your neighborhood.
Soon enough, smiths started to try some of the techniques used to make better bronze swords, to try and make less crappy iron ones. Most of these were still inferior to a really good bronze sword, but the raw materials were comparatively abundant. It quickly became apparent that ores from certain places produced better results, long before analytical metallurgy could say why. "Sky iron" from nickle-iron meteorites was particularly prized, and localities with particularly useful native alloys developed reputations. Some of the best attempts were arguably low-grade steel (carburized wrought iron), and were finally a match for bronze. As technology advanced, iron-using societies swept across Eurasia; societies without iron weapons and tools were marginalized, conquered, or destroyed.
So, in the most general terms, real-world weapon rankings might go something like this: copper < iron << bronze < complex iron? < complex bronze < primitive steel << good steel < complex steel. "Complex" in this comparison meaning significant layering and zonal composition differences for different functional parts of the weapon, created by a skilled smith.
Ready cultural knowledge of the secrets of steel, in times where it is otherwise unknown or at least a tightly-held secret of a few master craftsmen, is by itself enough to give dwarves both a reputation and an economic foundation as the "weaponmakers of the world". Knowledge of the techniques of magma workshops, allowing the entire chain of steel production to take dramatically less fuel, gives them a significant economic edge even when faced with human societies that have some knowledge of steel; and allows processing of "goblinite" to be practical. I'm looking forward to the economy improvements that will naturally put various societies and races into a more interesting relationship.