As far as I'm aware one problem is displaying text. Japanese readers generally need a Japanese font to correctly display Japanese text if it's Unicode. This becomes a problem when you potentially have text that can come from different languages. E.g. a Japanese font will display Chinese incorrectly.

On the web you can work around this using the lang attribute to tell the browser how text should be interpreted.

It's notable that, for example, traditional and simplified Chinese does not have this problem because they are encoded separately.

Another problem is missing characters. Some people have complained of not being able to write their own name. I'm not sure to what extent this has been solved through Unicode updates.