A sword is:
a convenient-to-wear sidearm, so easy to have with you if wearing it is legal and/or appropriate,
a sharp weapon, and thus much more effective as a weapon under many circumstances than blunt weapons such as maces,
often longer than a mace, and reach can matter a lot in fighting with weapons, and
often more agile than a mace. The combination of reach and speed provides a large advantage.
While a mace is potentially more effective against armoured opponents (but this will depend on the weight of the mace), note that armours typically had many gaps that could still let a sword be effective, and maces had plenty of competition as weapons more effective than swords against armour (such as polearms such as halberds and bills, guns, daggers).
But even if using a mace or any of these other weapons, it was still possible and often useful to wear a sword as a sidearm. Thus, on the late Medieval battlefield, even though armies primarily using spears and other polearms (including cavalry using the lance) or ranged weapons (guns, crossbows, bows), most of those soldiers also carried swords. The first point above - convenience as a sidearm - made swords very popular weapons even on a battlefield with plentiful armour.
Off the battlefield, one was less likely to fight armoured opponents, so the sword was popular off the battlefield, too.
For use against armoured opponents of the battlefield, the mace was convenient, but infantry would typically use spears or other polearms instead. This left the mace as a convenient weapon for armoured cavalry fighting other armoured cavalry. This wasn't close to enough to make the mace (and various warhammers, etc.) more popular than the sword.
In some times and places, maces saw wider use.
Maces could be relatively cheap (depending on the design), and didn't need steel. Cheap lightweight maces found use as tough weapons for the battlefield, and were used by those who couldn't afford swords, and could still serve as backup weapons in case swords broke or were dropped. Maces (and other clubs) were often popular in Stone Age societies where swords were not available.
Maces also saw use (in China) as police weapons (basically, as long metal truncheons).
Note that neither of these two cases depends on better anti-armour performance.
That ties in to how effective they were. Contrary to the popular belief that swords are literally useless against armour and that blunt force is the way to beat it, this doesn't reflect reality completely. If you take a look at manuals of the day that detail armoured fighting (pollaxe, longsword) from many different masters (Fiore, Talhoffer, Paulus Kal, etc) it is often seen that the blunt end of the axe (or the sword) is utilized mostly to create openings and/or threats to the person. The finishing move is more often than not the dagger of the pollaxe to the face, except for Kal who does like his chops. Even so usually just simple hits are relatively easy to counter, and the entire first chapter of Le Jeu de la Hache (The Play of the Axe) deals with different ways to counter such an attack
How does this translate to maces versus swords? Well, it shows that blunt force was not that great against the armour. While it can be dangerous if it hits, the if is the main part here as countering blunt hits is a lot easier than countering stabs from a pointy weapon. A stab is also a lot more deadly (a blunt hit may hive you a concussion, or it may require several hits to do so. A stab to the face is usually instantly deadly) and so the versarility of such weapons seems to be held in higher regard than the pure blunt-dealing damage of a mace. A harness, even one of full plate, still has weaknesses you can exploit with a pointy bit.
Late medieval armies, and let me remind you that this is the times in which a large portion of the army was equipped in half plate or brigandines in the vast majority of Europe, carried more swords than other sidearms. The situations in which they did not seems to relate to costs. Here's two examples down below:
The Sire of St Remy describes the english archers armoured thusly at Agincourt (1415):
"For the most part without armour except their pourpoints; their stockings rolled down to the knees, and having hatchets and battle-axes hanging at their waists, or long swords, some barefooted and bareheaded, others with huvettes or capelinas of cuir-bouilli, and others of osier, strengthened with a cross-band of iron."
Fast forward a few years, and you have this account from 1440 from James Skidmore (of English archers again):
“good jakks of defence, salades, swerds and sheves of xl arwes atte lest.”
And lastly one from 1483 from Dominic Mancini:
“There are hardly any without a sallet and none without bows and arrows…there hangs by the side of each a sword no less long than ours, but thick and heavy as well. The sword is always accompanied by an iron buckler…They do not wear any metal armor on their breast or or any other parts of the body except for the better sort…the common soldiery have more comfortable tunics that reach down below the loins and are stuffed with tow or some other soft material.”
You see that the hatchets and axes were used when the archers were lightly armoured, or not armoured at all, indicating it was a cost issue. When the archers are a lot better equipped in the subsequent accounts they carry swords.