In short, no, no sword of any type can cut or slice through plate armor, and piercing is exceedingly unlikely.

To see how two-handed swords were used when fighting in harness, we do have some, well, instruction manuals. With the earliest examples dating to around 1300 (for the M.S. i.33 "Walpurgis" manuscript showing unarmored sword-and-buckler fencing), and really hitting their stride in the 15th-16th centuries, we have a number of period fighting manuals. These are treatises, many illustrated, written by professional fencing masters for the benefit their students and patrons.

A number of these from the late 14th century onward include sections on fighting in armor with and against various weapons, often in a judicial duel context.

For our purposes, let's look at the examples given by Fiore Furlano de'i Liberi (c. 1340 - c. 1420) in his work "The Flower of Battle" dating to around 1404 and by Hans Talhoffer (c. 1410 - c. 1482) in his "fechtbuch" (lit. "fight book") from the mid 1440s.

Both of these gentlemen include a segment in their respective manuscripts a section on fighting in full armor with two-handed swords. In both cases, the response is not to go through the armor, but around the armor. They make extensive, almost exclusive, use of what the Germans like Talhoffer termed "Halbschwert" (lit. "half-swording"), where a combatant grips the blade of his own sword partway down its length. This essentially "chokes up" on the sword, and one would then use it more like a short spear or polearm to press and hook and maneuver the opponent and the weapon to guide the tip around the plate armor and into a more vulnerable point, such as the eyeslit, armpit, or groin. Especially in the German schools, a fight in armor seems to degenerate into an armored wrestling match (and it is no accident both Fiore and Talhoffer, as well as others, include extensive sections on wrestling). The Germans even called it "Ringen-am-Schwert" ("wrestling with swords").

Now, could a two-handed sword damage plate armor? Maybe, depending on what one means by "damage". There is a half-swording technique--Tallhoffer calls it the "Mordschlag" ("Murder stroke") where one grips the blade of the sword with both hands and, with a powerful overhead strike, uses the hilt of the sword like a hammerhead to bash an opponent on their head. Weather this would severely dent or damage the helm is going to be dependent on the helm, but it would certainly rattle if not stun the guy inside, allowing for followup techniques.

Now, on to the second part of your question: if a sword couldn't cut or pierce through plate armor, what use was it on the battlefield? Setting aside that most period "fighting manuals" seem to be more geared towards duels and judicial combat, one thing to consider is that, even at its height, full plate armor was not employed by everyone on the battlefield. Full plate harness is expensive, and was the purview of those rich enough to afford it. Many, if not the vast majority, of the enemies our hypothetical 15th-century knight is going to encounter on the battlefield are not going be wearing full plate.

Furthermore, you ask more specifically about a "Zweihander" type of two-handed sword. Now, what is and is not a "zweihander" vs a longsword is a distinction that largely only exists in our modern sensibilities. However, large two-handed swords most commonly associated with the Zweihander (also called a "Bidenhander" or "Dopplehander", all meaning the same thing--"two-hander") type hit their stride really mostly in the first half of the 1500s. They were used on the battlefield in an era where full plate was on its way out beyond ceremonial and tournament use. Formations of pikes supported by archers and firearms. Zweihander wielders fought along-side these pike blocks. They were particularly known to be used by certain members of influential mercenary companies particularly among the Landsknechts. (side note: popular histories will tell you that Doppelsöldners, those Landsknechts who received double-pay, used zweihanders. Whereas zweihander users were entitled double-pay and thus were doppelsöldners, not all doppelsöldners used zweihanders. A doppelsöldner was a volunteer who was placed in important and dangerous positions in the formation, such as the front line). It is said that zweihanders were used to disrupt enemy pike formations. Whether that was the case or not, as well as how they would go about that, is debatable.

You might be interested in some of the comments In this thread which briefly touch on the role of two-handed greatswords in pike combat. In short they're usually described played the same role as the other "short weapons", being placed somewhere in the middle of a pike square to be used in a close quarters and to defend the ensigns in case the formation in case the pike square ever lost cohesion and the battle devolved into "pell mell". They could also be used outside the pike square by certain officers like sergents for self defense. Or troops armed with short weapons could be mixed in with light skirmishers to back them up in a skirmish or to fight on rough terrain or assault trenches.

As for their use against armor, no swords wouldn't be able to actually cut through plate armor unless it was an extremely low quality. But despite the Italian wars possibly seeing the use of plate armor reach its peak many military writers still recommend two-handed weapons which can give powerful downward blows for use in armored melees. In particular there seems to have been some debate between shorter, stouter, halberds with wide, heavy, chopping blades which were considered better in a pell mell and long, lightweight, thrust-centric halberds which were considered better in a dual or a skirmish.

Two-handed swords were known for being able to decapitate and inflict other terrible wounds on unarmored targets, but against a man in armor a greatsword could still have enough force to disorient, concuss, or knock him to the ground with powerful blows.