I think I have a relatively unusual situation, so I would like to share it, if I may.
Although I live in the UK and have spent well over a half of my life here, I am Korean. Two of my best friends, whom I met at school in England, are Japanese and I speak all three languages fluently. I have been a FS fan for over 30 years but am passionate about many other sports.
Yes, there is no denying that there is a degree of deep-rooted animosity between the two countries. However, for most people, it is more of rivalry than hatred. We see it at work place, at school, in other sports - two top players ferociously competing but having not much against each other otherwise. Unfortunately, a small but significant sector in each nation do bear hatred and start sh*t-stirring whenever they see an opportunity, egging on others. Oh God, do the media love this! The rest of us range from disinterested to embarrassed. Do I root for Korea when they play against Japan in other sporting events? Of course I do - I'm Korean. Is it because I hate Japan? Nooo! I root for Korea regardless of who they play against, even if I know they are going to get absolutely thrashed! FS is different for me. My favorite female skater is Yuna at present, not because she is Korean but because I like her skating. My favourite male skater is Takahashi by miles. None of the pairs in recent years have stolen my heart because I still haven't got over Gordeeva/Grinkov.
Another factor is that FS is emotive as well as subjective to a degree. That, I think, is one of the reasons why I prefer this to, er, curling (apologies to curling fans out there), but it creates room for emotional arguments, more so than a match that was lost on a controversial foul call.
What is also significant in this situation is that both nations have fallen in love with the two girls before/rather than the sport. Perhaps it is not just these two countries, as far as FS is concerned; maybe we notice the individuals first, their countries next and then the sport. Not a very good example and not relevant to FS, but the following, I think, is a lovely story.
In men's 10000m speed skating, everyone KNEW the gold would go to Sven Kramer from Holland. Speed skating is HUGE in Holland, beyond most people's imagination. Kramer is unbelievably popular in his country. I mean, massive. He was skating last. With six pairs to go, this Korean skater, who had won silver in 5000m, is having a skate of his life. He actually overtakes the other skater's lap (unheard of at this level and quite embarrassing for the guy who also happens to be Dutch). Whenever this Korean boy skates past the Dutch crowd, do you know what they are doing? They are cheering him on, screaming their lungs out, because this guy is about to break the Olympic record, which he does. Then Kramer skates, makes the most ridiculous school-boy error (coach's fault!) and gets disqualified. He is devastated, the Korean guy gets the gold rather than silver (said afterwards he's happy about the gold but the victory feels incomplete because he won it through a stronger skater's mistake) and the Dutch crowd are still contratulating the medalists during the medal ceremony. They love the sport first and then their athletes who are good at it.
Sorry I'm rambling. It's just that there seem to be some misunderstandings and unduly strong emotions, occasionally irrational, on this board and I wanted to share my take on it for what it's worth.
However, if it had been a Korean who was beaten by Apolo, I somehow don't think the Korean contingent would be so overjoyed and supportive. Just saying.