The reasons for Japanese Korean estrangement are due to the nature of Japanese imperialism and how its colonial policies. Although your example of US/Britain is a bit of apples and oranges, but the comparison is instructive.

1. Culture: In the case of the Anglo-American world, there was still a great deal of cultural overlap among elites both pre- and post-independence. Although the US prided itself upon a frontier agrarian ethos such as Jacksonian democracy or the Jeffersonian yeomanry, the leading cultural institutions still hewed towards Anglo-Saxon models (such as university curriculum). This was in stark contrast to Japan, which instituted a harsh regime designed to eliminate or reduce an independent Korean culture. Koreans were third-class citizens within the Japanese empire (the Japanese were more tractable towards the Chinese on Taiwan). This cemented a long-held animosity in both countries. For Japan, the stereotype for Korea was backwardness and simplicity while the Korean popular memory emphasizes Japanese exploitation and chauvinism.

2. Economic colonial models : Although the British economic model for empire was predicated upon mercantilism (trade benefiting the mother country), substantial numbers of Americans profited from it. Whether in the plantations in the South or the New England cod fisheries and timber yards, the British imperial economy created a strong indigenous socioeconomic class in the Thirteen Colonies. Again, the contrast with Japan and Korea could not be more stark. The Japanese went about setting up an empire in a highly planned and deliberate fashion (as deliberate as empire building can be). This ensured that when Japan developed Korean agriculture and industry, it was to the benefit of Japanese. The peculiar nature of Meiji industrialization abetted this process as the Japanese state developed a very close relationship with the nation's private industry. This coordination meant that Korean development did not need an indigenous intermediary. The perniciousness of the imperial economy is evidenced by the fact that although the agricultural output of Korea went up under Japanese rule, the average intake of food went down for most Koreans.

3. Post-independence governments : In both Koreas, resistance to the Japanese became one of the central planks of their respective nationalist agendas. For the communist North, the Japanese were both imperialist exploiters, but also fascists by virtue of the Japanese alliance with Nazi Germany and its invasion of China. There is some historical debate about whether or not Kim Il-sung actually was a prominent guerrilla against the Japanese, but one thing that is clear is that North Korean propaganda lionizes Kim's anti-Japanese activity. Syngman Rhee also made much of his journalistic opposition to Japan and his activism to catapult himself into a prominent political position in the South after 1945. Both Koreas also try to stake claim onto the various Korean patriots and victims of Japanese colonialism. An Jung Gun is a prominent example of this phenomena. An was the assassin of Ito Hirobumi, one of the architects of the Meiji empire. Public propaganda and memory tries to claim both An as one of their own in the two Koreas. Kim Jong-il took a deeply personal interest in shooting a historical epic An Jung Gun Shoots Ito Hirobumi, which An is an unambiguous patriot (in stark contrast to the historical An who was very neurotic and whose motives were muddled). Similarly, South Korea has its own An memory cult, exemplified by novels like Looking for an Epitaph, a alternate-history dystopia in which An fails and Korea remains a part of the Japanese empire. The above shows that while both Korean states have a vested interest in keeping the memory of Japanese occupation alive. The US lacked this nationalist impetus against its former colonial master. Furthermore, anti-Japanese sentiment enjoys a degree of popularity in Korea. An has a tae kwondo move named in honor of him. And remember, the experiences of Japanese occation are within living memory for many Koreans. A Korean student of mine told me his grandmother speaks very stilted Korean because of the Japanese imperial language policy. The US demonstrates much the opposite phenomena. American leaders beyond the first generation of independence did not need to establish their anti-British credentials. Furthermore, Anglophilia has a degree of popularity in the US, see for example the recent fascination with royal weddings and births. Thus in Korea there are substantial pressures both from above and below to keep anti-Japanese sentiment alive. Such pressures are absent in the American case.

sources

Duus, Peter. The Abacus and the Sword The Japanese Penetration of Korea, 1895-1910. Berkeley: University of California Press, 1995.

Elliott, J. H. Empires of the Atlantic World: Britain and Spain in America, 1492-1830. New Haven: Yale University Press, 2006.

Myers, Ramon Hawley, Mark R. Peattie, and Jingzhi Zhen. The Japanese colonial empire, 1895-1945. Princeton, N.J.: Princeton University Press, 1984.

hi! FYI, you'll find some previous discussions on the US-UK relationship in the FAQ

Anglo-American Relations