There's no battle. Both Korea and Japan see the value in exporting cultural products, although I don't think the Korean government is focusing on webtoons as much as kpop.

Manga and webtoons are different and cater to different audiences.

In any case manga is much, much more popular and actually gets translations. I think this is due to the longer publishing schedules and how established manga is compared to webtoons.

Korean music is more popular in foreign countries than Japanese music, and their dramas may also be more popular, but manga, anime, light novels and Japanese games are so huge that there's no real competition going on here. I think many people overestimate the importance and popularity of Kpop.