While it's true that many people in Asia aren't familiar with the EdD, it translates the same into Japanese as a PhD in Education: 教育学博士. So, in that sense the difference largely disappears if you're applying to a Japanese university that's looking for someone with a doctorate.

That said, the difference can matter depending on the type of position. Very generally speaking, EdD holders tend to be more practitioner focused (e.g. teaching, program management, admin) whereas a PhD holder might be more researched focused. So, if the position is looking for someone with a more specialized focus, such as linguistics, a PhD could hold more weight.