You may want to read up on the history between those two countries. They have shared quite a bit...
Its been a few decades since I lived in Japan (Okinawa). And I have never visited Korea.
But I think that visiting both would be great fun. For planning purposes, I would probably advocate a historical/future type of approach. Japan has a long history I would consider climbing Mt. Fuji visiting some museums, some Shinto Temples, staying at a country Inn, as well as touring some of the electronic retailing areas. And I definitely would try to experience the food...
Is it Korea that has the most Internet Bandwidth available per capita in the world? I'd like to see what they are doing with all that bandwidth. Again, I'd be interested in some of the Historical attractions as well as the food...
Best of luck,
Uno