After the invention of plate, why did people continue using and, more importantly, romanticizing swords? Large, hard-to-carry ones seem like fairly unlikely candidates to become the symbol of medieval war when other weapons such as maces and spears were so much more effective, either because of their reach or against plate.
The same could be said for the katana of medieval Japan.
The most common battlefield weapons used by the samurai were the bow and spear (variants include certain yari and naginata, a polearm weapon), but almost never the katana. The naginata was essentially (in the broadest way) a pole with a katana stuck on top of it – if you could use the katana, you could use the naginata, only this time you were at up to (and sometimes exceeding) two metres away. Why use a sword when you can use a stick-sword? Katanas were almost exclusively a side-arm in battle (if taken at all, alongside the bow and spear), and then some more during the Sengoku period (a period of widespread civil war) where firearms were introduced.
The katana wasn’t truly the symbol of the samurai until the Edo period, an essentially two and a half centuries of unbroken peace following the Sengoku period. The katana was the samurai’s badge, his mark that told everyone that he was of a high rank, something that he carried with pride. However, even during the Edo, for self-defense many samurai still opted to go for swords such as the wakizashi or tanto, as they were a lot easier to handle in close-quarters than the katana – the katana was sort of the awkward middle-brother of swords, too short to have a decent range (in turn keeping the wielder out of danger), but too long to be used effectively as a personal-defense choice.
Alongside this, for many centuries already swordsmiths had been developing the technique of crafting the blade – it was during the Edo that smiths really exaggerated characteristics such as the hamon (the wavy line you see dividing the blade) and decorations of the blade to make them look beautiful beyond comparison, however this came with the swift elevation of their power, lifting katana to near-legendary status.
In terms of the spears or bow, and to answer the second part of your question - yes, spears (or naginata) were more effective, but were also uglier and unwieldy to use for ceremonial purposes. Also, imagine lugging a heavy spear around when you want to visit friends, as opposed to a relatively lighter and nicer looking blade.
The invention of plate armour didn't mean that everyone on a battlefield was wearing plate armour, and the creation of full plate armour is something that happened very late in the medieval period. A sword is quite arguably better than similarly sized weapons (axes, maces, morningstars) against lightly armoured opponents, due to its agility.
Also, swords really aren't that large or hard-to-carry, for the most part. The big "zweihanders" used by landsknechte and the like were exceptions. Most swords would fall between around 2-4 pounds, making them pretty lightweight - ideal for a weapon that is to be worn on one's side as, well, a sidearm. The image of a knight with a sword at his side is therefore a popular one - even if the sword wasn't actually the primary weapon, it is something that many men-at-arms would have had on them.
And, as a note, the effectiveness of the sword as a weapon didn't exactly end with the Middle Ages. The era in which full plate armour was widely used wasn't that long, really, and the sword managed to reemerge as a very effective cavalry weapon - something that would remain true all the way through the Napoleonic Wars. This continuity is something that certainly might have helped cement the sword as the symbol of warfare.
To draw a comparison to /u/bigbluepanda 's account for Japan, there is a well known quote by Aleksandr Nevskii of the then-powerful Kievan Rus:
'Those who come to us with a sword, will die from the sword'. There is a statue of him holding a classic medieval sword, and there are other prominent during the times of warriors using those classic large and straight swords(the largest female statue in the world stands in Russia, holding that sword).
The kazakhs and the infantry of Kievan Rus at the time did not use those swords a whole lot. Their more common weapons were axes, (cross)bows and Turkish-derived curved swords, a mix between the scimitar and a curved blade (sikh, IIRC). Why do all the images we still recall include that straight blade?
I believe it has to do with the presentation of the blade. It's a rugged, simple, heavy, long and straight blade, a symbol of the strength and resiliency of the people and countries of that era. Nevskii was a strong man, holding a strong blade, and the combination of the man and his weapon of choice has contributed to it being a symbol of power in the Slavic nations of the time.
Bonus fact: Nevskii's personal guard was incredibly effective at what they did, and quickly earned a frightening reputation for being powerful killing machines. Their weapon of choice? A long, straight sword, suited for combat on a horse.