The benefits of a flail over a sword certainly wouldn’t be in how quickly you could swing it, but in the ability of the flail to wrap around a defending object.
The thing maces were designed for was dealing with armor that was more or less impregnable to a sword. Some of the later sword designs that were meant to deal with heavy plate ended up being thicker and more blunt along the edges in favor of a long, thin, reinforced point that could be used against the relatively small gaps in armor. It got to the point where this approach wasn’t paying off very well. You ended up having to half-sword (grab the blade with your hand to reduce the length and increase leverage) and wrestle with your opponent to try to jab the point into joints or the eye-slit in the helm. Swords were still used because of the length advantage, prestige as a weapon, and because they were still quite useful against lighter-armored opponents than those wearing full plate.
While there were spiked and studded maces, most of them did not have long protrusions, the spikes were quite thick and relatively blunt. The main thing was to, like others said, provide a focal point for force, in the hopes of punching through armor. The spikes weren’t meant to cause tissue damage, though I’m sure that was a bonus if they did. Modern versions, meant to appeal to the fantasy crowd, are really different from quality reproductions or historical pieces, and the one I linked to there is about as historical as it looks. What you guys are calling a mace on a chain is more properly called a flail (this site also has a few other historically accurate reproductions of maces and flails). The weapon was derived from the tool.
The chains on flails were quite short; most were around 6 inches. They were not really used to get around a shield, but to increase the momentum of the ball while keeping the shaft relatively short, for better maneuverability and speed. Like a sling or atlatl, it was a way to increase the length of the lever to deliver more force. Another advantage was that you could ride by at full speed, whack a guy in the head, and not have to brace against the full combined force of your motion and your horse’s being transmitted to your body; it mostly goes into angular momentum on the flail head instead of being carried along the shaft directly into you.
More common were flanged maces, an example from the Arms and Armor page again, one at MyArmoury, with a surviving antique here, and some examples from a museum photo right at the top of the Wikipedia page. The spikes might look cool, but they probably didn’t work nearly as well as the flanges. Many historical maces I’ve seen pictures of (but annoyingly can’t find now) are a lot more like the Iberian mace example on the Arms and Armor page. That means that maces were designed around the same ideas as any other club: get as much mass out on the end of your shaft as possible while still being able to wield it effectively, and make that weight out of something as hard and durable as you can. Any other considerations are secondary.
Most o the dudes that own a mace/morning star actually own a modern, non-fighting, “designed to look”, cool wall hanger. I own a WWI Brit Trench Mace. It was made for killing the Hun" and it has short nubby “spikes”. It is very well balanced.
It was also not made for penetrating armor, which experts have claimed the Flanged Mace was for, as per Sleel.