I often hear things like how the Assyrians built their empire because they had iron weapons. I'm sure it's more complex than that, but is iron really that much better? I just watched this video and it looks like bronze's weakness can be annoying, but it doesn't seem like it would put you at a huge disadvantage.
Take a look at this thread, courtesy of /u/alriclofgar, and /u/WARitter. Steel is superior to bronze, but pure iron is inferior.
I think general consensus is that bronze was better than early wrought iron weapons up until the middle ages, but iron was much more availible.
Given that I would like to add as a side question why didn't some of the elites back then (roman preatorian guards, or early middle ages nobles) use bronze instead?