Ives also gives a good example of why one might want to change the key of a song. My reasons are:
1) The melody might sing better in a different register on the piano.
2) The overtone series of a different key can give it a different color.
3) I don't use identical voicings in every key. I tend to favor different voicings in each key. By changing the key, I force myself to use a different arrangement.
Here is Ives giving a demonstration of the overtones changing the color of the tune by moving the key up a half step.