It used to be that speaking Japanese in Korea was almost taboo. The history between the two countries is rocky--in the 20th century, Japan occupied Korea, and Koreans forcibly taken to Japan, as well as their children and succeeding generations-- were long treated as second class citizens. During the occupation of Korea from 1905 (Korea was annexed five years later) to the end of World War II, the Japanese ruled with an iron hand, and essentially tried to obliterate Korean culture (an ironic thing, since Korea has contributed richly to Japanese culture). Koreans were forced to learn Japanese, which certainly explains their later reluctance to use the language. There were many other atrocities, including the well known use of Korean "comfort women" during World War II. Tensions continue, in part because as with China, the Japanese have never really come clean on the atrocities they committed in the name of Imperialism--e.g., they continue to be glossed over in the textbooks studied by Japanese schoolchildren, and Japanese government officials continue to visit the Yasukuni Shrine, a symbol of Japanese military rule. But despite the official chilliness, on a personal level there are other stories that indicate that time has healed some of the wounds. An interpreter I met in Seoul told me that when she studied English in the United States, she roomed with a Japanese girl. They became good friends, and at one point, the Japanese girl broke down and apologized tearfully for all the things her country had done to Korea and its people. The interpreter told me "I could not accept such an apology--it is not mine to accept on behalf of all those who suffered", but that she was moved by the girl's feeling of remorse and her need to express it. They remain good friends and see each other from time to time.
Today many young people study Japanese freely and Japanese tourists in Korea (particularly housewives on shopping trips, since the exchange rate is very favorable) abound. At the airport, English, Korean and Japanese are prominent--it's Chinese that's missing. I interviewed one young man for a job who spoke extremely well. Despite studying for only a year, he was able to pass the Level 1 Proficiency exam offered by the Japanese government (I've passed Level 2, but not yet tackled Level 1). I was a little amused at how he did it, though--he confessed that he had fallen in love with the Japanese instructor, who was quite pretty, and was determined to pass the exam in order to impress her enough to go on a date with him. Sheepishly, he admitted that although he passed the exam, she ultimately rebuffed his advances.