The Greeks clearly had access to bronze from the archaic period and used large amounts of it to manufacture armor. Yet they still used iron for swords and spear points. This would indicate that iron was already superior to bronze for quite a few use cases.
Iron was mostly perfected by Romans through their conquests as they collected and systemised iron processing tech of the nations they conquered, which all entered Iron Age by then. By the times of early Roman Empire, high quality iron spread around enough to make bronze obsolete.
I think the main benefit of bronze and it still existed in Roman times is that you can cast it instead of forging. It took almost two thousand years to figure out how to make steel cuirasses that resembled the bronze ones Greeks used before the classical period. Of steel chainmail or scale armor was already superior anyway.