The main body of the japanese infantry was hardly heavily armored either. the issue had little to do with relative equipment as each side had their advantages (and one could in fact make a arguement that the Japanese were more or less at a disadvantage as they had virtually no cavalry units and also no artillery.) it was experience and organization. the Koreans had neither when the war started (those are two of the same thing really, it's hard to have organization when you have no experience)
The annals of Seonjo (the court records of the king) recorded many discussion of the Korean court during this war, in one passage, they talked about how their archers both out range the arbeques and was in fact more accurate, the King then ask why they were losing so terriblly, and the answer was that because even while being more accurate they still could only do limited damage in a distance, and when the Japanese closed in the Korean lines simply collapsed. neglecting whatever advantage their bows can bring. as the Japanese were totally use to taking a little casualty from arrow showers at a distance, but the Koreans paniced and fled at the first signs of significant casaulties.
Just how inexperienced were they? before the war started one of the head Korean general reportedly said that they'll have no problem with the Japanese because "our men are bigger and stronger"
yes, a head general of a military in the late 16th century thinks that the key point of warfare is that you have big strong men, and not you know.. actual organized fighting. When the Japanese marched to the most northern corners of Korea and met against the the elite forces of Korea , it's northern guards, the Korean cavalry actually did a number on the Japanese in the open field. but then the experienced Japanese simply regrouped in a nearby hill and set up a defensive formation, the Koreans took no regard to this and simply tried to charge up the hill and fight them. the end result wasn't too hard to predict.
The Koreans also wasted away their army in the field in the early going . when it was pretty clear that their makeup would have favored them much better if they were to just try to defend the city (you know, where you can force more individual fightings so your "bigger stronger men" might actually do something. and your archers have more time to fire off their shots), meanwhile the end result was them wasting the army in the field and then fleeing the cities without a fight.