I've heard a couple battle videos with a different music from the rest. What does that signify, and how can it be used?
If you're talking about the in-game WiFi battles no music has been changed.
Are you watching them on Youtube?
If so, the most likely reason why your hearing different music is because you're listening to a replay with edited music. That means they owner of the video can play any music they want over the top of their battle.