I hear / read that it would be in both Japan's and South Korea's (SK for short) interest to be friends and/or allies. It's been acknowledged by both gov' and the USA wants it too (I think).
So why are they still not so friends ? Every military exercises where Japan and SK have to be together, they are actually "exercising with the USA", but not really together. And the list goes on of small stuff that could be improved on their diplomacy.
I hear it's because Japan did not make formal apology to SK for its past WW2 crimes, nor made any form of memory and introspection work on that period.
Why is Japan not doing it ? Why is Japan not working bigger on their relationship with SK when it's in their biggest interest in any matter (diplomatic, military, economic and whatnot) ?
Japan simply thinks they have apologized enough. They call their current situation "apology fatigue". Basically the people are tired of feeling like they have to keep apologizing for something that happend "so long ago". Any Japanese politician who would issue another formal apology would commit political suicide. They would lose the support of their own party and of a big part of the population. This isn't just in regards to South Korea but towards all of Japans neighbors.
So for now Japanese politicians who want the two countries to work together try to find wiggle room. Japanese politicians generally acknoweledge that the South Korean people went through very hard times but do so without admiting Japanese fault.