Medieval swords, contrary to popular belief were not the massive beams of steel that many think they were, and Rapiers were not as light as many assume.
On average a long-sword and a rapier both weighed about 3.5 pounds, with the main difference between them being the distribution of weight. Rapiers had lots of hiltwork and long-swords had broader blades but it was the way in which they were handled that made a difference. The rapier had the skinnied-out blade, a development of the real battlefield sword having to put more force into thrusts, and, as such, it was a stabbing weapon*. The long-sword was used for slashing attacks against its contemporary adversaries, who could be promptly diced with little armor for protection.
By the time gunpowder was starting to play a major role in warfare (ca. 1630 C.E.) many extra-thick breastplates were made for protection. If you ever see a small indentation in a museum piece, that was the test shot at the armorer's shop.
*In fact, some rapiers were not sharpened at all along their sides.