The North American version of Sonic CD not only changes its opening & ending theme songs, but also changes the whole soundtrack! (Except the Past themes, which were played in PCM) Ever since late 2011, versions of Sonic CD with both the Japanese and American soundtrack became available worldwide, with the former lacking lyrics in the opening & ending themes due to being unable to clear the rights with the singer's estate.
Destiny, Eternia, Symphonia (and sequel), Phantasia (GBA and iOS) and Legendia all had its vocal theme songs replaced by original orchestral compositions. Symphonia (and sequel)'s and Legendia's compositions are special in that they were actually composed by series composer Motoi Sakuraba for the international release. The GBA version of Phantasia simply replaced the song with its overworld music, while the iOS version used an orchestral song synced to the video of the PS1 opening FMV.
Thief II: The Metal Age provides a peculiar example in its German (and only German) localization. The credits video for the game and a few ambients in certain in-game locations use the song "Accingite Vos", by the German band Subway to Sally. Weirdly, all other versions of the game don't include it, to the point that even in the English language version, there isno background music in the credits video. The lenghth of the credits video and how well it syncs with the song seems to indicate it was originally planned to be released as part of the non-German versions too (including the original English language one), but some last-minute copyright issue probably prevented that. Some surviving online news releases from the time of the game's development indicate this might have been the case, as there was no mention of the band contributing music only to the German language version of the game.
The English intro to Yo-kai Watch uses an original theme by Jeff "Swampy" Marsh (co-creator of Phineas and Ferb) instead of the translated version of "Gera Gera Po" (which was used for the early episodes of the dubbed anime before being replaced with the new theme). Yokai Watch 2 features a remixed version of the English theme rather Japanese versions remixed "Gera Gera Po". The intro's visuals are also different.
The English version of Monster Rancher 4 used two new songs by The Cooler Kids for its opening and ending themes instead of the original Japanese songs or translations of them.