I am a blacksmith with metallurgy education and make swords as trained by a reproductionist smith who aims to make period-accurate weapons. I also like to read about sword history. Here is my answer.
Here's the thing: any modern steel is likely to outperform the "steel" of the 13th and 14th century. Even using the most basic modern steel grades (anything from the 10XX series, I typically use 1080) would, strictly in terms of mechanical properties, put 13th-14th century steel to shame. Any industrially produced Steel is going to have a multitude of small additional metal additives to perfect the alloys performance. Even if we were to make a batch of Steel from scratch that was just iron and carbon, our ability to control the carbon content of our metal, remove impurities from it and have fine control of the temperature of the heat treatment is so advanced that we can control most of the mechanical properties to a degree unheard of even 150 years ago.
Now, let's think about steel in the 13th and 14th centuries, because here's the thing: it is barely really steel.
While bloomery furnaces had been around for centuries at that point, these early metal refining practices could only produce wrought iron, and the Smith would have to go through an elaborate process of combining multiple low-carbon irons and carborizing them (adding carbon to the iron to make steel, but not so much it becomes cast-iron), folding the metal to homogenized the final metal, OR by adding low carbon iron into a carbon charged furnace. While this would produce a steel, it would have a varying range of carbon content and an incredibly low degree of control over the final carbon content of the Alloy, and hence its performance. The creation of the blast furnace an the oxidizing environment it provides is needed to really start producing true Steel, and it wouldn't become a common production method until the 14th century, with the first known example i believe being found in the 12th century. Hell, England didn't get a blast furnace until the late 1400's.
What I'm getting at here is that the production methods of the 13th and 14th century were just so outmatched by our current metal production standards that even a sword made by a master smith of the time would just not hold up to one made of modern metal in terms of mechanical properties. The average metal produced at the time just was inferior.
Now granted, even with amazing metal if you were to give it to an amateur Smith of the era they could still easily produce an absolute piece of crap if they don't properly heat treat the blade. Here's where your question becomes more difficult to answer because when you ask "how do they fare against each other" I'm assuming you mean "in a fight." See, even then it would be hard to really evaluate the effectiveness of a modern sword versus a peroid weapon due to all the variables of who made it, who's using it, where was it made and with what methods, AND how well have they been maintained, etc. Really the performance properties of swords that we'd be evaluating here for "which is better" really arent going to make a huge difference in just ONE fight. The primary characteristics that you'd want the sword to have would be enough toughness to not break in a swing but enough hardness to maintain an edge on contact, but not so much the blade becomes brittle and cracks. In combat these properties aren't really competing against each other, sword versus sword. People weren't clashing their swords together and having them break on contact in combat: typically they would at worst Bend, have the edges chip or cracks form if you were to actually see a failure, and a low-quality metal sword would quickly lose and deform its Edge after a few swings, but even a blunt sword was a deadly weapon for its ability to just shatter bones beneath a heavy swing. Regardless, it would still take a high degree of abuse to truly break a quality-made sword, historical or modern.
So here is what i can say with confidence: if you were to take a sword made today with modern material and modern methods and take it to a lab and compare its mechanical properties in terms of bending resistance and edge hardness/maintenance, the modern one wins always. Average european smiths wouldn't have access to the furnaces required to get the heat needed to produce quality steel even remotely comparable. In terms of their combat performance the period-Made sword could likely approach the performance quality of a modern sword with enough time, skill, and resources, but it would quickly degrade under use much faster than a modern blade would and the modern blade would maintain Peak Performance quality for longer than the standard 13-14th century blade. The historical blade will lose/chip its edge against the modern blade if u clash them together, so it will certainly lose in a war of attrition against the modern blade, but that rarely was the conditions of a sword fight. Honestly if you had two swords in front of you preparing for a fight, either one, modern or historical, would serve you just fine unless it was made by someone incompetent, but over multiple battles take the modern sword if you want to fight with a better weapon for longer.
If a modern Smith used every available technique and scientific advancement we have today to create a sword as a replica of a historical version, I have no doubt that blade would stay sharper longer, be less likely to chip, bend, crack, or rust, and in general have properties comically beyond what is actually required for the sword to do its job of cutting a thing to death, but a historical sword can also cut you to death well enough that you won't notice the difference.
Some books on my shelves l looked at and that will inform you better on the techniques of the times:
H. R. Schubert, History of the British Iron and Steel Industry (1957).
Alan R. Williams, Methods of manufacture of swords in medieval Europe, Gladius 1977
The 14th century is when all-steel swords start to become common (but still in a minority). For the 13th and 14th centuries together, perhaps about 1/4 of swords were all-steel, with the remainder being steel-iron laminates, with either a steel skin around an iron core, or steel edges on an iron core. Many of the all-steel swords are made by welding multiple pieces of steel together. This trend continues through the 15th and 16th centuries, with all-steel becoming somewhat more common (perhaps 1/3 of swords in the 15th, and 1/2 in the 16th). All-steel, from a single piece of steel, becomes the main method of construction in the 17th century.
The steel itself varies from, for all practical purposes, as good as modern steel (similar to modern 1060, 1070, or 1080 carbon steel, but with more slag inclusions) through to low carbon steel that couldn't be usefully hardened by quenching.
There are two big differences between modern swords and swords of that time:
Modern heat treatment of steel can be done consistently (if we know the composition of the steel, we know the optimum heat treatment, and we have thermometers and thermostatically-controlled ovens). This doesn't necessarily affect quenching so much, but it has a large effect on tempering. But not being able to temper reliably and consistently can mean that quenching is done conservatively or even omitted. As a result, 13/14th century swords have hardnesses all over the place, from about 200VPH to 600VPH (12HRC to 55HRC). The median is about 300-350VPH (30-35HRC).
Modern swords often have terrible geometry and poor construction. This is especially the case for (a) cheap replicas and (b) fantasy swords. They can start with good low-slag steel of known composition that a Medieval swordsmith would have been most pleased to have, and make clunky badly-handling inferior weapons. In the worst cases, swords that cannot reasonably be considered to be functional weapons. This results from swordsmiths who don't know how swords are used, from making swords tough enough to cope with modern abuse (users who try to chop down trees with them, etc.), swords for modern re-enactment fighting, and ignorance of the swords they are trying to make replicas of (e.g., working from a photo and measurement of the length, with no measurements of the blade thickness).
So, how would these old swords far against modern swords? If we take the best old swords (which are metallurgically similar to good modern swords) and the best modern swords (which replicate the old swords in terms of weight and handling), there would be very little difference. If we we to take medium or poor quality swords, the modern ones will be better metallurgically, but the old ones will usually be better weapons in terms of weight, balance, and handling. Especially with one-handed swords used with shields or bucklers, you don't get much sword-on-sword contact (compared to modern fencing and movie and stage swordsmanship), so being metallurgically inferior doesn't matter that much. The old sword is likely to be a better weapon in practice than the modern clunky sword.
Some past discussion on the strength and toughness, modern vs old, mostly by u/angry-mustache and myself: https://www.reddit.com/r/AskHistorians/comments/9x06v7/how_does_the_average_steel_weapon_of_the_middle/
Reference:
Williams, Alan. Sword and the Crucible: A History of the Metallurgy of European Swords up to the 16th Century, BRILL, 2012.