Personally, I don't think weapons need to be "balanced," until and unless you're talking about fantasy weapons, like the Dwarven Urgosh, or those quicksilver swords.
RW weapons evolved for different reasons and for different purposes. They got used by different people. There are real world reasons why one person would chose a longsword over a axe, or vice versa.
Part of the problem, of course, is that the weapons and armor on the D&D equipment charts come from a lot of different cultures and developed over a lot of time. By the time the rapier became popular, the longsword and halberds were largely ceremonial weapons, and 2 handed axes were rare as hen's teeth- as was plate armor.
But D&D doesn't (and never has) accurately modeled the various weapons anyway. There was an article I read comparing the lethality of thrusting weapons (like rapiers) to slashing weapons (like longswords and katanas). Researchers compared the wounds found on various warriors bodies and experiments on body gels, and even examined the historical records of duels between Eastern and Western warriors when the West was opening up trade with Eastern powers. The conclusion was thrusting weapons were more lethal, but slashing weapons had more "stopping power." That is, much like a bullet, a slashing weapon sends hydrostatic shockwaves of energy throughout the body, and a single strike might disable a warrior from shock without actually killing him. Warriors fighting with slashing weapons tended to have multiple healed wounds and a lot of deep scar tissue. However, thrusting weapons penetrated several organs at once- rapier duelists who lost tended to have several wounds, any of which would be considered fatal, but, because of the lack of hydrostatic shock, didn't drop the opponent quickly. Because of this, they fought on after they were fatally wounded.
Ummmm...I guess that's my longwinded, tangential way of saying: Leave the weapons alone, and let players choose them for roleplay reasons.