I think alot of this is geting too politically correct on one end and then totally biased on the other.
The problems are a result of how vague the question is. There are people who are talking about armies now and time periods and all these other details because there was nothing specific about it mentioned.
But if youre gonna look at armies then keep in mind that Asian warfare (I'm gonna mean Chinese here) was about almost a blitz. First a huge rain of arrows to weaken the ranks of attacking armies then a storm of cavalry to break up the remaining regiments followed by a horde of infantry to pick off stragglers. This was all to be surgical and sufficient without retaliation. Basically meaning that it was a defensive blitzkrieg (yeah i know it doesn't make sense in english).
Compare that to the more straight and regimented attack force of western military it'd be about how good the
shields are and how long can the asian army last if it can't breach the defense. however since we are talking recurve bows and platemail of the period chances are the arrows would pierce and the cavalry would trample but you never know.
If we go all out European and Asian then really the idea of Longsword or (i'm more familiar with the broadsword argument) Katana (actually the odachi if were going with longsword) is kinda lame. Full out war east and west probably means the entire fleet. never mind nations and allegances so you end up with different pairs.
since going about it full army wise is just making a mess we should go about it in those separate pairs almost like an exhibition.
For brute strength you have the Norwegian Beserkers with the war axes against the huns with the lance or halberd (not going for the horseback archery here otherwise the Beserker would be slaughered and that would ruin the point) Both forces have been known to wreak havoc across the southern reaches of Europe. I'd say it would come down to luck. Both are aggressive and focus on quick bursts of offensive thrusts and charges and both have remarkable endurance.
For cavalry you'd have the zanmato wielding samurai on horseback with recurve bow for range against the shield and lance of a period european horseman. In this situation the tactics would be key and the asians are known for their cavalry tactics. Both are used in their armies for shock and to scatter regiments but the zanmato and other asian variants are meant specifically to damage other cavalry (after all if you fight in east asia its a must) Wether or not the zanmatos would immediately disable the enemy horse the shock value would present and the following distancing would let the samurai or relative warrior use their bow to capitalize.
For straight infantry you could go with the standard knight with broadsword or axe versus the samurai wielding standard set of dachi, kozuka, wakizashi, and spare odachi. This match would still be a tie the reason being that the
chainmail and plate
mail armor is waaay too thick to pierce with any glancing sliding or what have you attack a samurai's weapons would be capable of. Hence the archers again. But likewise the heavy and cumbersome european armor and heavy broadsword make the agility of any attacks quite easy for the much lighter samurai to dodge. basically you end up with a scene of the samurai using everything but the kitchen sink (and then later the kitchen sink) to make any useful strike on the knight before the blades end up dulling while the knight keeps lunging for the samurai or trying to catch the katanas in a lock or trying to push with a shoulder thrust and not gettin anywhere. Then it would come down to endurance and really now i think both would collapse before any advantage is gained. Luck might help but really the samurai would have to be pretty poorly trained to take a bad hit from the knight and no knight is gonna be stupid enough to leave himself open to one samurai.
For close range finesse fighting we go to the obvious European rapier and Chinese jian. Both are not meant to take heavy strikes with a key focus on dodging and parrying to unbalance a foe before capitalizing. In the case of a fencer a lunge. For the jian a break to get close enough for a quick cut or jab into a weak point. Both armors would be kept light so as to avoid restricting movement but thick enough to prevent glancing blows. This usually means a basket guard and leather pads for the fencer and leather strap armor for the jian swordsman. now i might be a fan of the jet li swordfight in fearless but super cool high jumping wushu moves aside the fight would probably be similar to that. A fierce and fast fencer keeping enough distance from the swordsman to anticipate openings which would be just what the swordsman needs to break form and catch a weakness in the epee to get close for a killing blow. typically the jian would be used as little as possible holding off the rapier when absolutely necessary but not enough to mess with the edges. That usually also means blocking with the flat side of the blade and working with the impacts. Unlike the movie it would really come down to how keen the fencer is with keeping up variety in the attacks and the swordsmen's versatility with the weapon. If the fencer can keep the swordsmen at a distance, length of battle may win it for him as it takes much less energy to lunge with a rapier than it does to use a jian effectively. BUT because the jian is very effective the swordsman just needs to make sure he catches all the possible openings and even one moderate one can be a kill shot. It is much easier to parry with the jian and its free handle than work with the basket guard on the rapier. Older forms would still be less balanced and the fencer still needs a firm grip on the handle unlike the swordsman with the jian.
For mid range infantry you come to the polearm or halberd and the Qiang or Da Dao. Speed is key for singles. Regimenting is key for groups. Da Daos are usually used in singles but those weapons would tear through halberds and polearms, more of a match for pole axes but at their range and speed they would screw up an axe wielder. Qiang groups and spear groups would be better matched and may keep at a standstill. (Again archers) (actually maybe even cavalry) On the european side though you have to go with their tight formations with shield reinforcement. The tougher regime would break right through the light phalanx of the eastern armies and pick off stragglers. Just the groups i see a complete European domination and that's probably the better way to go about it.
For long range its all about the arrows and again the East are known for their archery. Multiple crossbow and recurve style defeats western archery every time which is also why we ended up adopting that form too. And then started using guns. And then RPGs and incendiaries.
Next up is style. on the contrary to what the asker said, there is actually lots of variation to styles and weapons on both sides. The difference is that (well also cuz it became more Japan vs Europe than Asian vs Europe) the Japanese modified what they thought was the perfect formula which they perfected from the Chinese Dao and proved every time that it was definitely superior to the dao. but even then you have the odachi, dachi, kozuka, wakizashi, kunai and a few others that aren't as mainstream. moving out of Japan you get the jian (your straight double edge) a few kinds of axes and lances, the da dao and halberd, the butterfly swords, the hook swords, and even shuriken which are basically throwable knives that were redesigned to fly sorta better. And also contrary, western broadswords weren't all or even mostly double edged. Many broadswords, arming swords, and long swords were cruciform but sharpened on only one side. You also have angled swords and single/double sided axes and who can forget the smithing genius of the hollow tip javelin? These styles were all built from combat style (in the east) and war experience (in the west) to be better suited for the type of battle waged.
The samurai typically wants fast and easy to continue circular slices so the slicing power and balance of the angled and single edged dachi made sense. The heavier claymores and zweihänders of the west provided better strikes and lunges for knights. I still love the agility of the rapier and versatility of the jian tbh. Nothing says I'm gonna destroy you like somebody throwing tons of strikes at you within seconds while keeping you pinned.
Wow that was long lol but hey i had 3 pages to work with.