People often compare sword with handguns. Handguns nowadays are deadly weapons, but they are not used in war, where people use assault rifles instead, or other heavier weapons. Swords were sidearms. Warriors carried them around every day, they could be used for self defense (in those times there was not police or law enforcement). Or they could be used for intimidating opponents, and if needed, for chopping someone's head or limb.

In combat though, spears were much, much more effective.

With one exception. As far as I can tell only one exception, but a big one. The Romans have figured out that all other people are quite bad at close quarters combat. Since soldiers were used to using spears, very few soldiers knew what to do if someone got in a close range than the length of a spear. The Romans then would bull-rush the enemy using their shield, and when close they would use their (quite short) swords. Without training, their opponents didn't stand a chance. Of course, this type of battle tactic required an enormous amount of training. The Romans did that because they were a very war-oriented society. Others couldn't match their level of training.

Even for the Romans though, who were regularly using swords, the choice of bronze versus iron would have made little difference. Although bronze is a bit more brittle than iron, and a bronze sword is more likely to break when it strikes a hard object, the Romans were trained to thrust the sword, not to use slashing moves. Thrusting is much more deadly, but less instinctive. It's another thing that requires extensive training, which the Romans were great at. For thrusting, I think a bronze and an iron sword do equally well.

Nice explanation, but the Romans did use spears - the 7-foot heavy pilum. They just threw them. It was effectively used to disrupt a threat in a mass volley before swords were drawn and the zerg rugh. Several battles were won by the pilum - the battle of Telamon for example.