Ace Combat 3 - PS1. The Japanese version is far superior. I remember when the Japanese version came out it was immediately sold here in Macedonia, we bought it with my cousin and thought it was incredible. We didn't understand the story but we couldn't stop playing it and finished it. We couldn't wait for the English version to understand the story....all of it was removed. The first shock came when Official Playstation Magazine UK gave the game we loved so much 1/10, then we realized what was going on.
The English release was definitely not 1/10, far from it, I dunno what happened there, but it wasn't the Ace Combat 3 we knew and loved.
Astal - Saturn
I have a beef with this one, it's not just because my first experience with Astal was the Japanese version, but because it's objectively superior.
The opening movie that plays on the title screen actually has an amazing song in the Japanese version.
The game is also better balanced, it has a selectable number of lives and I could be wrong on this but better health balance too.
Ninja Gaiden III - NES/FC
I don't know the exact differences, as the only version I've known is the Japanese one, that's the one that was sold in Macedonia usually, but from what I understand it's much easier and better balanced.
TNNT III: The Manhattan Project - NES/FC
In Japan this one is known as TMNT II and it was often the version sold in my country, it had a nice Options menu where you could set the game's difficulty and the number of lives. On the US version, this options menu was removed and hidden behind a cheat input, just awful.