Some classic and modern writers have attempted to portray the evolution of swordsmanship as a linear progression from worse to better, or primitive to advanced- presenting various technical principles, like the supposed superiority of the thrust over the cut, as objective advances. Some also present developments in sword-making as advances in technology from something obviously worse to obviously better- lauding the theoretically more agile and defensively superior 18th-century smallsword or court sword over the larger, supposedly clumsier rapier of the 16th and 17th centuries, for example.

In some cases, this framework is correct. Good steel is a more or less objectively better material to make a sword out of than bronze, for instance- tending to be both harder and tougher, holding a keener edge for a longer period of time and being able to stand up to more abuse before bending or snapping. In other cases, it sort of makes sense in a nuanced way- Iron Age swords prior to getting proper steel figured out were kind of a sidegrade to Bronze Age weapons in terms of quality, since iron is harder/edge-holdier but more brittle than bronze, but they still represented a substantial advance from the point of view of a society because iron was less expensive to mine and work.

Modern consensus on historical fencing, though, views thinking of the history of swordsmanship in terms of a straightforward progression from worse to better as being the wrong way to go about things. Instead, fencing changed in reaction to how swords and swordsmanship fit into the broader cultural, technological, and military context of the time.

Very generally speaking, swords and swordsmanship in the 1400s were influenced by the growing adoption of plate armor: the one-handed cruciform knightly sword, designed to be used alongside a shield, was giving way to narrower, larger two-handed longswords because a larger weapon is better at dealing with armor, and because improved armor made shields less necessary for defense. The sword at the time was still largely a battlefield weapon, something used as a noble's sidearm in open warfare- a lance or polearm of some sort being more likely to be used as a primary weapon.

Swordsmanship in the 1700s was strongly influenced by the growing dominance of gunpowder warfare. Armor was much less common, and while swords in general were still in use as a battlefield weapon in certain contexts- like cavalry sabers- it was far more common for the sword to be used as a weapon in civilian self-defense or dueling, because while the sword was becoming militarily obsolete, owning and carrying something like a rapier or smallsword was still expected of the upper class at the time. So, naturally, unarmored civilian self-defense or dueling was the kind of thing that contemporary teachers of fencing tend to emphasize.

Basically, I would expect the median swordsman of 1480 to beat the brakes off of the median swordsman of 1750 in a fight in full armor with arming swords or longswords, and I would expect the reverse in an unarmored duel with rapiers or smallswords.

This is not how swordsmanship works.

For one, 15th and 18th century are meaningless without also giving us a place. 15th century Indian swords are quite different from 15th century French swords. Not having read the novel you're referring to, I'll give a broadly European answer, but this eurocentrism is problematic in and of itself. It should also be noted that an alien world, even if it never developed beyond a medieval european level of technology, would likely still have developed different weapon designs and martial arts to go with them. Still, to further illustrate my point:

Swords, the techniques they're used with, the situations they're intended for and the other equipment, especially armour, all exist in an interrelated system. As such, the sword techniques from the 18th century aren't "better or "worse" than the ones from the 15th century, they're intended, designed, developed and adapted for the environment of the 18th century, which is quite different. There is no linear evolution towards aa constant "better" than before, merely an adaptation to changing circumstances.

A famous swordsman and duelist from the 15th century might for example use a longsword, for which highly developed systems of swordsmanship existed, of which we know most about the German school based on masters such as Liechtenauer and the Italian school based on masters like Fiore to only name the most well-known to us today. At this time, duels might have been fought unarmored ("Blossfechten"), but techniques also existed to fight against full-plate armour ("Harnischfechten"), which would have been worn by knights, men-at-arms and nobles on the battlefield from the late 15th century through the early 16th. Their main battlefield weapon would have been the lance.

A swordsman of the 18th century would likely either use a smallsword, which is optimised for unarmored duels against other smallsword-users and the comfort of everyday wearing, or if they're more militarily inclined a sabre or backsword, although that all depends on location as well. The main battlefield weapon at this time would have been a firearm or a pike (when in formation). Armour was far less prevalent.

A smallsword would be at a huge disadvantage against the longer and heavier longsword. Even though smallswords are capable of very quick thrusts, the longer range of the longsword negates this advantage and the very light smallsword would have a hard time parrying longsword blows. It is also pretty useless against armour. Smallswords are optimised for high society honour duels and were used in a time when even on the battlefield armour wasn't as prevalent anymore. Needless to say, the techniques for a smallsword are completely different than for a longsword.

A sabre or a backsword, both being heavier than a smallsword and capable of both cutting and thrusting compared to the smallsword, which can barely cut at all, as well as having decent hand protection, isn't as helpless. In my personal opinion the odds still favour the greater reach of the longsword, but a more skilled sabre or backsword user would have a chance in an unarmored duel.

Of course, this all presumes that the person using 18th century techniques has access to 18th century weapons. The stated scenario makes this unlikely. As such, the combatant with an education in 18th century techniques who has to fight in a 15th century environment with its weapons, armour and conditions, would actually be at a great disadvantage, as their techniques wouldn't work with the weapons at hand.

I would say the question is moot because of the amount of time between the two compared eras. If we consider the form of sword that each respective era would likely have used then we understand quickly that the technique used to wield a given sword was radically different. The skillset that your Anton uses to win duels is likely using a light sword meant for single combat - likely a rapier or estoc or similar weapon. However, if the setting this book takes place in is to be the late 15th century, then our hero and his rapier are in peril. The armor of this period is completely primed to make his weapon and technique completely useless. The rise of the longbow and later the increased use of black powder weapons would eventually cause armor to become less and less common, as encounters such as the Battle of Agincourt show us that the concept of heavily armored knights was being challenged.

So lets back up a bit and say our Anton challenges a noble to a duel - that noble shows up for the duel in a full plate rig with a long sword. Can Anton reasonably hurt this man? No. Can that man reasonably hurt Anton? Yes. Can Anton's skill with an 18th century sword overcome these differences? Maybe, but probably not. We have to assume that the hero of this book has no experience fighting in extremely heavy armor, as this was falling well out of favor. This protagonist would have perhaps gone into battle with a breast plate and helmet, but not much more. Fighting in full armor was in itself a skill, and that skill is not something that the time traveling "nobleman" would have possessed. I thus think it is unrealistic that such a man would be a great swordman in this setting and be able to say that the source of his greatness is his 18th century swordsmanship training. The skillset needed to flourish as a fighter in the 15th century would have simply been completely different because the challenges posed to the swordsman (armor) was completely different.

In short: Yes there were massive differences between swordsmanship in the 18th century and the 15th century. The advantages the character possesses would not do him many favors in the 15th century unless he was fighting an unarmored opponent, which a nobleman would certainly not be. The character could of course adapt to the time period and don armor/learn how to fight with a weapon that could defeat armor, but then it would not be logical to say that the source of his success as a swordsman comes from his 18th century fighting technique. Cheers!