With the advent of plate on the medieval battlefield, both the design and techniques of swords began to shift to handle the new threats. Two-handed weapons became more popular as the shield was discarded. Swords often became longer and thinner (with a few exceptions, such as the famous sword of Henry V). You can see a few examples of these longer blades in the category of swords that Ewart Oakeshotte labeled the Type XVIIIs. When fighting against other armored knights, a man-at-arms would use a technique called half-swording, where he would grip the sword by the middle of the blade in order to thrust through the gaps of his opponent's armor. You can see the man on the left going for a thrust, while his opponent is employing the mordschlag, where you take the blade in both hands and bash your enemy over the head with the hilt of your sword. Every part of the sword is dangerous in the hands of a proper swordsman.

Maces were always a common sight on the battlefield, but knights began to adopt the poleaxe in increasing numbers. There's a wide variety of different types and names for these, but essentially they're something of an all-in-one polearm. A poleaxe might have an axe-blade on one side, a hammer on the other, and a spike at the top. Some had more of a "beak" type of blade for penetrating armor. These were enormously popular weapons for soldiers in the late middle ages. The long shaft allows for a two-handed swing, therefore giving you more power to penetrate or crush through an opponent's armor. Since men-at-arms generally didn't have any use for a shield, the one-handed, much shorter mace was less useful for them.

Even though the poleaxe was arguably more popular for fighting against armored men while dismounted, it did not render the sword irrelevant in warfare. Medieval knights carried multiple weapons and used each according to the situation they found themselves in. A mounted knight used his lance in the charge, then switched to sword, mace or axe if his lance broke or if the charge was bogged down in enemy infantry. If ordered to dismount (or forcibly dismounted by an archer) he would have waded into melee combat with a poleaxe and a sheathed sword and dagger. Weaponry was never an either-or decision for a knight.

I cannot speak directly to the Middle Ages of England, but I can speak about a similar development that took place during the Bronze Age of the ancient Near East. Up until the early 1200's B.C.E., the stone mace was the best weapon a warrior could carry. It could easily smash through whatever type of armor a solider wore into battle.

During the Iron Age, however, the surrounding peoples learned how to work with iron to a much better extent. Iron had already been largely used during the Bronze Age (the terms "Bronze Age" and "Iron Age" were developed during Europe's industrial revolution and are misnomers, more reflective of the ideals of 19th c. Britain, France, and Germany). But during the Iron Age, many peoples, especially one group of sea-fearers known as the Philistines, began crafting a majority of armor from iron. Iron was a much tougher metal than bronze and required a substantial amount of force to break. Because of this, other weapons, such as spears and battle axes, became more widely used.

Based on this example, we see that weapons often adapt to new types of defensive technology and vice-versa. A solid mace in the hands of a strong enough opponent could rattle an enemy and possibly dent plate armor, but it would not break the armor. Swords, on the other hand, can still hit vital parts of the body not completely covered by armor. These places, as displayed in crusader era British armor, are areas that required flexibility (i.e.: joints). A sword had a must better chance to pierce a weak place in the armor than a mace had in causing it to break.

The excavation reports of Tel Megiddo and Tel Hazor deal with the development of weaponry in response to defense, which is a pattern that we still at play today.

In addition to the great extensive answers already in the thread, let's remember that at no point in the Middle Ages were most participants on a given battlefield equipped with all of the latest technologies. There weren't any battles (at least on a large scale) in which everybody was equipped with full plate, and in most battles, the majority of participants weren't necessarily even knights. So, yet another reason for swords to continue to be used after the advent of plate armor is that a) most people didn't wear it, and b) those that did didn't necessarily have it on all parts of their body.

Thank you all for the great answers!