I am now going to compare a sword a European sword against a katana. Who would win and why?
Yeah, I know, I'm pissing on the electric fence here. That's the fun of the game, isn't it?
Before I compare, let me go through a few scenarios.
Let's compare a master swordsman with another master swordsman. Who would win at that skill level? That's anybody's guess. These guys are masters at their craft. They earned this title through years of practice. At that level. you should't be able to guess. Neither of the men in the fight are pushovers. You learn nothing general from this scenario other than which master swordsman could beat the other.
Let's compare novice swordsman against novice swordsman. Who would win at this skill level? That's anybody's guess. These are two very inexperienced swordsmen making mistakes as they fight. You can draw no conclusions where novices are involved.
This leaves us with comparing average swordsmen to average swordsmen. Anyone who mentions as "master" after this point volunteers to be ridiculed. We compare average to average as this will produce a the most reliable comparison. Neither warrior will be acting stupid, and neither warrior will be acting extraordinary. These will just be two well trained humans fighting.
I will begin by comparing a Charlemagne era man-at-arms (800 CE) vs. a Japanese warrior from the height of Japanese swordsmanship. Who would win? My bet is up in the air. The European has a shield, and in case you don't know, shields are awesome and save your life. In addition, the European wears chain armor, which would be excellent against the slashing attacks of the katana. Meanwhile, I think that the Japanese armor of this period is generally superior to the European armor. All in all, I have to call this a wash.
Could the Japanese swordsman have cut through the chain? I doubt it. If that had been true, then the Europeans would have used slashing weapons more often. Instead, they stuck to the straight sword, which indicated that the straight sword was more useful. It's not like they didn't know about the curved weapons from out east since Charlemagne fought and defeated Moors in Spain. In fact, given Charlemagne's general success in war, you must conclude that the weapons and armors of that age were reasonably effective.
The second reason that I doubt that Japanese swordsmen cut through chain is that the Japanese used chain armor. If it wasn't effective, why did they wear it? That sort of armor is quite expensive, so the expense must be worth the effort. In other words, chain must have been effective against the katana because the Japanese clearly thought so, and if Japanese chain is effective against the katana, then European chain must have been effective as well.
Past the 800's, the balance of power shifts towards the Europeans. As their chain became increasingly effective, the Europeans abandoned slashing weapons, opting instead for increasingly large and pointy weapons. If we trust their trial and error, then we must conclude that the most effective weapons against this increasingly good armor were a variety of brutal weapons, none of which looked like a katana.
By the time that you reach the full chain mail body suit, circa 1100, the average European warrior would be far more likely to defeat the average Japanese warrior simply because he could make more mistakes and not die for those mistakes. By the introduction of the tin-can knight, the katana no longer resembled a threat. The average European warrior had the tools and the training to handle tin-can warriors, while the average Japanese swordsman did not. In short, the finest swords slashing swords in the world were outclassed by the finest armor in the world. Defeating this armor too both specialized weapons and specialized training. Note that there is nothing special about this training. Given time, the Japanese could have adapted their own fighting styles to serve the same purpose.
The Japanese knew that western armor was the best armor. If they could get a hold of pieces to wear, they did, because they weren't stupid. This stuff outclassed everybody. If there's anything to learn, it's that anyone who faces death at the hands of an enemy likes the best armor that they can afford.
Note that I am not saying that the Japanese warriors or the European warriors were better or worse. What I am saying is that advancing armor technology gave the Europeans an increasingly wide edge over the Japanese. This advantage came from the fact that this armor was highly effective. Western armor simply outclassed the threat of the Japanese arsenal.