Korean Christianity in the 1900s succeeded, in part, because it failed so spectacularly in Japan in the 1600s. The answer, in my mind, thus has two parts.

The failure in Japan

Japan today has a fascinating relationship with Christianity - Christian-style weddings and cross necklaces and biblical imagery and so forth are everywhere in Japan, but basically no one is Christian as we Westerners understand it. The first Jesuit missionaries showed up in Japan from Portugal in the 1500s, and because their iconography and rituals looked a lot like Buddhism, and they were happy to sell guns to Japanese warlords, the religion was actually pretty successful for the first 50 years. By 1600, important samurai warlords were converted Christians and Catholic infrastructure was becoming a permanent part of Japan. Nagasaki, the second a-bomb target, was initially founded by the Jesuits and retains a legacy of being the capital of Japanese Christianity.

The big change came around 1600 when William Adams, an explorer and pirate, shipwrecked in Japan. He was not only the first Englishman in the country, but also the first Protestant. He convinced the new Tokugawa shogunate that the Catholics were up to no good, pointing to the horrors they were inflicting on Native Americans in Central and South America, and warning that Japan was next on the menu. Those familiar with James Clavell's historical fiction novel "Shogun" will immediately recognize that almost all of that story is true and based on Adams' life.

The Tokugawa shogun (Ieyasu) was already skeptical and jealous of the growing influence of the foreign religion, just as his predecessor Oda Nobunaga had been hostile to the powerful Buddhist temples, and Adams' offer to supply European goods without Catholic strings attached was too good to refuse. Over the next thirty years, the official Japanese relationship with Christianity transformed from support, to antipathy, to outright persecution.

Eventually a state-sponsored Inquisition was formed to exterminate Christianity from Japan, and it worked. Catholic priests were captured and tortured to death, or forced to publicly renounce the religion. Converted Japanese either left the country, apostatized or went into hiding. In fact, there's still some Christians in Japan today who have been practicing a version of the religion that's been underground since the 1650s (the kakure, or "Hidden Christians").

When the Spanish sent a delegation to try to convince the Tokugawa to re-open relations, the samurai beheaded all the diplomats but one, who was sent back to the Spanish with the message that if they ever set foot in Japan again, or even if God Himself shows up, the samurai will behead them. Since that point, Christianity has never successfully found its way back into Japan.

This is not for lack of trying - after the 1860s, missionaries were allowed back into the country with the wave of Westernization, but few Japanese converted. Even after World War II, when American missionaries were flooding into the country with much-needed food and medicine during the Occupation, no one converted. I've met a few former missionaries (Mormon, Baptist) who say they basically their time in Japan trying to convert people was totally wasted.

My understanding is that the 300 years of illegality made the religion a permanent Other for most Japanese. It's this weird foreign thing that's slightly subversive and, among the young, kind of cool. Oddly, Catholic schools run by the Church are pretty common in Japan, and the non-Catholic Japanese are happy to send their kids there because of their reputation for quality education. But it's not a religion "real" Japanese people do.

The success in Korea

Over on the peninsula, though, Christianity is flourishing. About 30% of South Korea is now practicing some form of Christianity, esp. the Korea-native Unification Church. They are basically like the Americans of the East, sending out Korean missionaries abroad and arguing about censoring scientific geology and evolution from science textbooks.

One of the biggest explanations for why Korea is so Christian is actually the Japanese occupation of the peninsula. After they annexed Korea in the late 19th and early 20th century, Japan imposed strict prohibitions on Korean cultural expression. The Japanese language and writing system became mandatory in Korean schools during the occupation, and eventually the occupation authorities banned Korean and hangul from most public use.

Of course, the Japanese by this point had a long and antagonistic relationship to Western Christianity. Because of geopolitical situation, though, one form of nationalist expression the Japanese were reluctant to censor was Korean Christianity.

Korean Christianity became a beacon of the resistance, a symbol of something that the Japanese invaders were not, and moreover something they were afraid of. Indeed, many of the pre-war leaders and organizers against the Japanese were strongly associated with the Korean church, and at the end of WWII this victorious group was the basis for the proliferation of homegrown Korean Christianity, e.g. Sun Myung Moon's family.

So in Japan, Christianity was a symbol of Western imperialism and remained unpopular, while ironically in Korea it became the nexus of national resistance against Japanese imperialism.

EDIT for few mistakes

Hooray! One of my favorite topics! I've studied this extensively since it's part of my own family's history.

First of all, there are already some posters insinuating that Christian missionaries experienced nothing but smooth sailing in Korea compared to China and Japan. This is absolutely false. Roman Catholicism was introduced into Korea in 1603 and was subsequently outlawed in 1758 as a subversive and dangerous religion. Christians were heavily persecuted since the religion's introduction into Korea. This ultimately culminated into a full fledged pogrom when 8,000 Christians were slaughtered in the Catholic Persecution of 1866.

I could speak on why Christianity was never able to gain significant popularity in China and Japan compared to Korea but that is another huge post of its own. There would also be a lot more qualified people to talk about that, but I'd say I'm one of the few people here who have a good background of Korean history and especially Christian history in East Asia.

I mean if you want to get really precise, the actual history of Christianity in Korea begins with Nestorian Christians who came by way of the Tang Dynasty over a thousand years ago, but for the sake of brevity, we'll keep it relatively early.

So if Catholicism was nearly stamped out, what makes Christianity so popular in Korea? Well the first thing to point out is that Christianity isn't nearly as popular in Korea as most people think. 46.5% of the population is irreligious. That's way more than China's irreligious population which is only at 12.6% -- most of China is dominated by ancestral worship (56.2%) and folk religion (12.6%). If we lump together Protestants and Catholics, only 29.2% of Koreans are Christians.

Historically, the northern part of the peninsula was populated by the literati and intelligentsia of Korea. One of the primary ways that European Christians were able to enter into non-Western nations was through education. Thus, the north freely welcomed Protestant missionaries who brought Western education, hospitals, and schools. The missionaries brought a non-traditional form of education that was enthusiastically embraced by proponents of the Silhak school. Silhak reformers emphasized merit over social status and were often from the middle class. They were able to afford good education but had no hope of aspiring high because of their low birth.

The children of the northern Korean middle class who went to these schools also studied abroad. They lived in Paris, New York, Berlin, Moscow, and London. They forged deep friendships with Western elites and were exposed to new ways of thinking: assumptions about human nature, different forms of government, the legitimacy of rulers, concepts of nationalism and national identity.

Unsurprisingly, these students returned as staunch patriots who despised Japan's encroaching colonialism. Herein we see the second factor for why Christianity was relatively well received in Korea: it was closely associated with the mothers and fathers of the Korean independence movement.

Third, Korean Christianity was a largely indigenous movement. Most of the European missionaries who attempted to gain significant footholds in Korea failed. Instead, it was the native Korean laity that kept the momentum of the religion going. Those who wished to be leaders in the community studied in seminaries and schools abroad and returned to minister to their own people. After the Empire of Japan annexed Korea in 1910, there was little to no interest from Europeans to go minister there. Korea was regarded as a backwater whereas China with its enormous population and Japan with its high level of industrialization were more promising prospects for new believers.

The fourth is because of Hangul, the native Korean writing system. Jeong In-Ji, a scholar in the Joseon Court, once said, "A wise man can learn [hangul] in one morning. A fool can learn it in ten days." Hangul is one of the great marvels of linguistic achievement. It's extremely easy to learn and virtually impossible to misread because there are few grammatical exceptions. This meant that anyone in Korean society was able to read Christian literature.

The final factor is societal change. Christianity was used as a justification to spearhead massive changes in Korean society. Minjung theology could be seen as an example of liberation theology, an interpretation of Christianity that empowers people as self-determiners of their own destiny even in the face of seemingly impossible hardship and persecution. That was a very appealing message to Koreans. Residents of the peninsula have long suffered under the invasions and ambitions of the two powers that sandwich them and vied for strategic control of Korea.

Christians also challenged the power of the Korean emperor, who they saw as a man who was subject under the authority of God, no different from anyone else. They challenged the existing social system which conferred women and children no rights. Jesus similarly challenged the social norms of his time which dismissed any woman or child's testimony in a court of law. Through the teachings of Paul, they also criticized polygamy, concubinage, and forbade nobles from deserting their wives for being barren.

That, in a very small nutshell, is why there are a good chunk of Christians in Korea. A lot of that population is concentrated into megachurches, but as a culture, Koreans have a tendency to take things to extremes. The mindset is that you are either 100% committed to something or there's no point, and that is not simply confined to service, but you are also expected to completely conform to the church's theology, leadership, and protocols.

Claiming that Christianity grew in Korea simply because it as seen as anti-Japanese ignores and dismisses a lot of factors. After all, Kim Gu, who is revered in South Korea as the Father of the Korean Republic, didn't become Christian because it was seen as a resistance religion or a symbol of independence against the Japanese. Very few Korean Christians did. He became a Christian after his fiancée died in 1903. Most revolutionaries came from the north because they had studied abroad and were inspired by other indigenous resistance movements. They were able to do that because they received a Protestant education and a lot of them became Christians. The religion wasn't seen as a symbol of Korean independence by the natives until years later. But at the same time, Korean pastors during the annexation period certainly preached sermons drawing comparisons between Koreans and the Israelites in Egypt.

Sources

A History of Christian Churches in Korea by Kyoung Mae Bin

Sourcebook of Korean Civilization: Volume Two by Peter H. Lee

Christianity in Korea by Robert E. Buswell Jr. and Timothy S. Lee

Christian missionaries were initially very successful in Japan, but unlike Korea, Japan had a major period of anti-Catholic policies, beginning in 1587. Prior to the policy changes, and the subsequent executions of Christians, the number of Christians in Japan was estimated at ~200,000, or about 2% of the population. The Tokugawa Shogunate stalled and then reversed this trend under penalty of death; these policy measures were enforced, and Christianity successfully suppressed, until the 19th century. The religion never really recovered.