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In Japan, players noted that the US version had three extra bosses including Sephiroth, Kurt Zisa and Ice Titan, and since developers wanted to give Japanese gamers the chance at the extra features without adding just three boss fights, they made several more improvements to "finalize" Kingdom Hearts; hence the Final Mix in the title.
In Final Mix, there are new heartless, new weapons, and new abilities. In addition, Heartless have been recolored, the entire game is in English with Japanese subtitles, and several new scenes were added to clarify certain plot points (these were not voice-acted, however).
Final Mix also featured a crucial plot point that would carry over to Kingdom Hearts II: the first meeting of Sora and Xemnas, the leader of Organization XIII. At the time of Final Mix's release, he was known simply as Enigmatic Man. This was an optional boss fight that could be initiated in the Castle Chapel area of Hollow Bastion.
An additional secret movie was added as reward for completing Expert Mode or completing the game 100% on Normal. Titled "Another Side, Another Story: Deep Dive", it delved a bit farther into the now-famous Roxas and Riku fight on Memory's Skyscraper.