I've had this question posed many times and the answer is not an easy this is better this is not.
firstly let me say a sword of any kind is a "tool" and is only as good as its wielder.
That being said, anyone who says that the "katana" is a superior weapon is blowing smoke, because the truth is neither is superior to one another. An experienced practitioner in the Katana can easily kill an in-experienced practitioner of a longsword. and vice versa.
to adequately explain how the weapon works you have to look at three things. Where the weapon is from, what the weapon is made from, and what the weapon is designed to do.
Firstly The Katana is From Japan, a nation which was Cloistered from the rest of the world for a very long time. With this in mind you have to know that for nearly all the time the Katana was used, the Japanese only really made war with each other. There were never any wars with other nations until the french (i believe it was the french, but i could be wrong on this one) came sailing off Japans shore and fired their cannons across one of japans major cities at this time. Thats right folks, gunpowder scared the bejeezus out of japan so much that they opened their ports to we savages and started trading cause they'd be damned if they werent going to get some of that black powder.
The longsword by contrast was European, a continent filled with nations wanting to kill one another. so the longsword or if you go back far enough, the arming sword (single handed sword) or even the gladius, was a weapon meant to wage war on many nations. No xenophobia here, were either going to conquer you or enslave you etc etc.
comparatively the katana was made to fight kill and conquer the japanese by the japanese. The longsword was made to fight kill and conquer "the rest of the known world" Advantage? Longsword (at least in my opinion)
Now on to what the Katana is made of versus the longsword.
Firstly calling a Katana a "Marvel of Metallurgy" is probably accurate simply because at the time in feudal Japan the type of steel they had was, for lack of a better word, Crap. much of the time it was powdered or recovered in small chunks and the molecular structure just wouldn't hold together very well. so the sword makers came up with a solution (that was the marvel) By folding the steel they could remove some of the carbon from the metal and give themselves one very fine solid edge that was sharp enough to cleave through a man. but thats it. one edge. and the drawback was that the blade was so brittle on the flat that you could shatter it like glass. Now keep in mind these are the blades from way back in the day. if you were to buy a Hanwei practical plus Katana made today. you would easily have a sword better than any Samurai had back in Feudal Japan. except for maybe the Shogun himself.
By contrast the longsword metal is solid and easily forged into a sturdy blade which can easily hold two edges. The blade would flex to allow blocks and parries on the flat of the blade as well as the edge.
comparison: Katana - fine edge, brittle and breakable on the flat, low quality steel
Longsword - more flexable, able to hold 2 edges, high grade carbon steel
advantage: Longsword (thats two)
Used for:
this is the big one, remember how i said that the Katana was designed by Japanese to kill Japanese? this is where the fighting styles evolved. now Killing is Killing regardless of what implement you use, but each and every Katana style is specifically designed to use a Katana, and to defend against a Katana. (Keep in mind that this statement is based on the fact that at the time the Japanese had no experience fighting other nations). This is a Slashing and cutting weapon and weighs about 3lbs, your cuts with this weapon were drawn in a slicing motion much of the time to make use of the katanas fine edge this would do the most damage to a target.
The longsword by contrast was used to make war on, well lets be realistic, EVERYONE. it also weighs about, you you guessed it, 3 lbs. because it was a straight blade and not curved you could have maybe an inch more reach with it if not more. The blade is designed for Chopping and Hewing (impact with precussive force causing a cleaving effect) slicing the edge like a Katana is at best an annoyance. because of these characteristics the longsword fighting styles would incorporate front and back edge cuts in addition to thrusting (something a Katana to my knowledge cannot do even remotely as effectively)
now these are just the characteristics of the weapon.
remember the MOST IMPORTANT RULE with ANY sword. is the training, skill level, and abilities of the person Wielding the weapon.
Thanks