I downloaded the Japanese version of PSIII, and I've been playing the English a few days before which I also downloaded (not exactly the same as playing it on a consple, but still fun). There are two noticeable differences, one which I already knew about:
In the English version of the game, on the overworld, only clouds scroll. In the Japanese version, the ground scrolls too. In dungeons, the backgrounds also scroll. This is mentioned at the cutting room floor: https://tcrf.net/Phantasy_Star_III:_Gen ... ifferences
The other difference is that in the English version of the game, when a character is poisoned it shows a P symbol next to their name. In the Japanese version, their HP is replaced with a P symbol, and this makes poison even more threatening.
I think there are a few other differences too, but I can't confirm them. If you do know any, feel free to tell them here.