The premise of your question is wrong. Japan absolutely did come under the influence of Western religion and culture, thereby integrating itself into the world economy and Western geopolitical landscape long before Korea entered the game.

The first real contact either Japan and Korea had with the power of the West was at the middle of the 16th Century. Antonion da Mota, a Portuguese trader, was blown off course and landed on the Japanese island of Tanegashima in 1542 or 1543. He showed off his arquebuses to the Japanese, and taught them how to load and fire them. The weapons fascinated the local Daimyo, who purchased some for his own and practiced with them until he reportedly became a marksman. From Tanegashima, handheld firearms would spread like wildfire, often using the vector of Portuguese (and Spanish, but mainly Portuguese) missionaries who were eager to spread Christianity to Japan. And if that involved buying the local lord's loyalty with weaponry and power, then praise Jesus.

The Jesuits and Capuchin societies competed fiercely for the loyalty and souls of the Japanese. While the lower classes seemed the most eager to convert (Christianity had a comparatively egalitarian message compared to the highly stratified systems of Buddhism, Confucianism, and the divine lineage of Shintoism) the nobility was naturally more influential. Many lords, especially in the south where the Europeans had easier access, converted and granted relatively unlimited access to their peasantry to the Missionaries.

All of this activity - Christianity and gunpowder - fueled a whole new class of lower-class light infantry called the ashigaru developed, and changed the face of Japanese warfare. Previously, Japanese warfare was based around the samurai, low-level nobility, who competed with a rigid (yet brittle) code of honor. The ashigaru, being mostly formed of the peasant class, had no such restrictions. Their primary weapon was the arquebus (the samurai considered firearms to be a coward's weapon and so almost exclusively preferred their katanas or naginatas at best). This massed firepower is what propelled Oda Nobunaga from a minor Daimyo to the supreme Lord of Japan in control of both the Shogun and the Emperor. His successor, Hideyoshi, inherited this army structure, and used them to launch an invasion of Korea in 1592.

One more note about these armies: there were only four gunsmithies in all of Japan. Unlike in European armies where a soldier who owned a gun might have gotten it from one of the hundreds of gun smiths and be grouped into a heterogenous unit of dissimilar firearms, these Japanese lords of war organized their ashigaru units so that they all shared arquebuses from the same gunsmiths. This ensured that the guns and ammunition were made from the same mold. I.e. if you shared ammunition on the battlefield, you knew that your comrade in arms had the same weapon as you. Whereas were you a European soldier, if you ran out or lost your ammunition, your comrade might not have the same shape of bullet, and your very expensive, very powerful arquebus is now just a heavy club.

The Korean's, by contrast, had pretty much no interaction with the West by this time. Joseon Korea, a vassal of Ming China, had pretty tight control over who and what came into the Kingdom, and constantly referred to China, whom they viewed as their "older brother" for assurance, assistance, and outright permission. In exchange, they received a lot of economic trade and support under the Chinese Tributary System. Korea was China's most reliable vassal (allegedly) and was consistently counted on.

While Japanese society was rather heterogenous, Korean society was comparatively homogenous. In medieval Japan, one might wander through societies ruled by major or minor clans, urban Imperial centers, and theocratic Buddhist warrior statelets. In Korea, there was one king, and below him were a class of hereditary nobility, and a bureaucratic class of Confucian gentlemen. This Confucian rule over Korea was a near total takeover starting from 1392. The Kim dynasty of Goryeo was still loyal to the Mongols (many many family ties, as per the Great Khaan's plan) and was considering shifting their loyalties to the Northern Yuan and helping them invade China. A lieutenant of the Goryeo, the future King Taejo, launched a rebellion, ousted the Kim, and established his own dynasty: the Yi. Since the Korean Buddhist sangha were partly to blame, and seen as corrupt and decadent, the Yi set a whole bunch of restrictions on the Buddhists, banishing them from the cities, seizing many of their properties, placing them among slaves and prostitutes as the lowest class, and then raised up Confucian bureaucrats on the Ming model in their place.

We'll come back to Buddhism later.

In 1592, the Hideyoshi's samurai invaded Korea. There's a whole religious angle to this war: Hideyoshi put Christian samurai and Christian ashigaru at the vanguard of the invasion with the (prospectively) goal that they would be among the highest casualty rate, and thereby thin their numbers back in Japan. Some of these newly Christian samurai clashed with the older Buddhist/Shinto lineages during the war, competing over land, glory, and etc. Many of the Christians, and this invasion is when the first known European, a Portuguese missionary, ever set foot on the Peninsula, viewed this as a holy war, to not only take Korea, but to convert it to Christianity.

The Koreans had little idea of this, and fought like mad men before the Japanese bullets. The Korean army, which saw little action, crumpled, and much of its navy was burned at harbor, or scuttled by cowardly or inept commanders. Only by overstretching the Japanese supply line, guerilla warfare from the locals, a massive Chinese counter attack, and the naval genius of Admiral Yi Sun Shin were the Japanese finally forced to abandon their Korean invasion in 1598 and launched a full retreat. (This is the nutshell-edition of the Imjin War.)

In the meanwhile, Hideyoshi - an old man at this point, who would die before his war would end - wound up with an international incident in the form of a Spanish treasure galleon from the Philippines beaching on his shores. By law, anything that touched the soil of Japan was officially the property of Japan, so Hideyoshi was eager to claim the ship and its treasures as his own. This treasure galleon formed a full quarter of the Spanish yearly income, so the Spanish were keen not to lose out on it. The Captain was taken to Hideyoshi and presented a globe to him. He showed Hideyoshi how Spain controlled not only Spain and the Philippines, but half of Europe, and most of North and South America. Hideyoshi gazed at the globe and wondered, How can such a small country rule so many others?

The Captain - in what I'm sure he thought was a genius move - told Hideyoshi that was the genius of the Spanish empire: first they sent the missionaries to convert the populace, then they sent the conquistadors once the people were already Christian. Hideyoshi did the math in his head: Christians had been running around his empire for years... and now this man was threatening him with a Spanish invasion. Hideyoshi flipped his shit and executed as many Christians in Japan as possible, resulting in one of the most famous martyrdoms in Asian Christian history: the 26 Martyrs of Japan.

After Hideyoshi died and the war ended, Christianity was soon driven out of Japan in a one-two punch of the Shimabara Rebellion and the policy of sakoku (closed country). Tokugawa Ieyasu, somewhat ironically, constructed his new Japan along the Confucian lines of the Ming and Joseon.

In Korea, the war had such a traumatic effect on the society, culture, economy, and political system that I dare say the country didn't recover until after the Korean War of the 1950s. The most obvious noticeable change was that Korea became even more restrictive over who was let into the country. Nearly the only trade was now with China (briefly to the Manchu, who then took over China anyway). There was a single office left open in Busan, with exclusive trading rights with the lords of Takeshima, who were vassals of Japan. But they were guarded day-and-night and had no freedom of movement in Korea. Korean commerce was even more strictly controlled by the Confucian bureaucrats, with linguistic education in only Chinese, Japanese, and Manchu.

cont'd