Japan couldn't promote their pop culture as heavily as Korea because of the people's fears of Japan becoming another military and political power. Asia's fear (especially those who were colonized and experienced their atrocities) of Japan returning to their WWII selves posed as a huge impediment in advancing Japan's pop culture. In other words, Japan's colonial past became a barrier for their immediate expansion, following their rise to an economic power during the 1970's~1980's.
South Korea has no historical imperial past, so countries usually perceive them as a "good" country without any strings attached. They wouldn't be scared if ever it becomes an economic power since they have no indications in the past that they would like to become an Asian hegemon. Because of this, they would allow Korea to continuously (and I might add, vigorously) campaign and promote in their country. They can easily broadcast their own channels such as Arirang, KBS and tvN, ensuring and strengthening the Hallyu wave.
On the other hand, Japan's pop culture expansion or promotion is generally perceived by their previously colonized countries as a form of cultural imperialism. That's why they can't easily broadcast their own entertainment channels to back up the growing fascination to Japanese pop culture, unlike what Korea's doing. If ever Japan was able to broadcast their entertainment channels and increase their promotion overseas, I don't think the Hallyu wave actually even stands a chance.
It's sad because they're actually the ones who started it in Asia. They started their promotion of pop culture WAY BACK in 1970's. However, due to their past atrocities influencing the people to think that they still have plans of becoming a hegemon, they couldn't fully utilize it. It's just.....sad.
*Hegemon: a state who is economically, militarily and politically powerful. Synonymous to what the US was after the Cold War period, around 1980's. US to some extent is still a hegemon due to its economic and political power, but China's emergence and various events have lead to their decline.
*Also, if you're wondering what book I'm reading for my thesis, it's this one: Iwabuchi, K. 2002. Recentering globalization: Popular culture and Japanese transnationalism, Duke University Press, USA. It's quite a good read, and it says a LOT about Japanese Pop Culture. A background on Behavioral Sciences would be good though to fully understand this book, I think :)