One of the biggest differences between Japanese and American animation is that in Japan, voices are recorded after the animation is completed and the Japanese actors do their best to fit the animator's work. The traditional "Disney-way" of animation records the voice performance of the actor first and the animators animate following the energy given by the professional voice actors.
The result is an acting performance which is arguably more convincing. The animator has the freedom to animate characters actually articulate what the character is expressing or emoting thusly creating a greater sense of believability. There are few Japanese animation directors who animate in this style, "Akira"'s Katsuhiro Otomo being one of them.
When Disney films come to Japan they are often retitled. "Tangled" retitled (?) to simply "The Tower of Rapunzel"(とうのうえのラプンツェル) and "Frozen" was retitled to "Anna and the Snow Queen" (アナと雪の女王)
Titles you can get away with tweaking to fit your audience, but dubbing the voices isn't as easy. Meanings can be completely lost easily and the clash of a foreign language on top of another performance from a completely different language can look and sound awkward.
In the case of "Frozen"'s "Let It Go", even if one could translate the meaning exactly to Japanese, it's highly unlikely the mouth articulation would match in English. To the credit of the Japanese director and translator they were able to find the best match for the Japanese translation to the English mouth articulations. So "Let It Go" becomes "Ari No Mama De" ありのままで which roughly translates into "Be As You Are", but the title "Let It Go" remains the same. (Performed by Takako Matsu. A Japanese translation can be found here.)
In the past, films Disney films with song numbers in Japan would simply be played in English with Japanese subtitles below, but the rest of the film would be played out with a Japanese voice-over.
To Disney's credit, during it's heyday during the 1990's translations were made for other countries way beforehand. For example, the songs for Disney's Tarzan was distributed in over 30 different countries with Phil Collins doing versions in German, Spanish and French. (The Japanese version was done by Masayuki Sakamoto) In this case they were lucky as the songs were played over the action montages of the film. No loincloth song and dance number for Tarzan.
