Broadswords and Rapiers seem to have appeared fairly close to the same time period. How did they fare against each other?

by solepsis
PopeOnABomb

Follow-up question:

Were rapiers more intended for dueling (thrusting) than general battle? (which leads me to wonder how much dueling and field combat would have had in common.)

If so, would it have been likely to find these weapons facing off? I am specifically asking about a scenario where both adversaries have properly prepared for the type of combat about to commence. The thought that led me here is why wouldn't one bring the weapon most well suited for the encounter. (why bring a dueling sword to a battle or vice versa?)