Dirge of Cerberus: Final Fantasy VII
Dirge of Cerberus: Final Fantasy VII
When being translated for international release, the development team made several drastic changes to counter the negative reception the game received in Japan. These included allowing Vincent to use attacks while in the air, being able to double-jump, and running 1.2x the speed he originally ran. Additionally, the camera angle in which the game was perceived was changed, in order to make navigation easier.
Contributed by Dazz
In the Japanese release, the game featured an online multiplayer mode with several story based missions, which even included cutscenes. This was removed as part of the game being reworked for the western market, though several of the multiplayer missions were made available as extra single player missions, without any story elements.

The International Version, later released in Japan, retains all the changes made to the western releases, but also includes the cutscenes from the multiplayer mode.
Contributed by Dazz
The opening cutscene of Dirge of Cerberus explains why Vincent and Yuffie weren't present in the final battle with Sephiroth at the end of Final Fantasy VII (besides them being optional characters). They were both in Midgar helping evacuate the city at the time of the final battle with Sephiroth.
Contributed by Pogue-Mahone