The iOS game adds a "Quest Medley" mode, which puts you through a random pairing of "field" and "battle" levels, different each time you play. It also adds some kind of composition mode, along with the ability to share creations via Twitter.
Unlike the 3DS version, the iOS release is free to download. As reported by NeoGAF users, it comes with just two songs, with the rest purchased as in-app bundles. Even additional characters are sold via IAP. In fact, user Goli notes that, when the NZ release of the game is opened in a US account, the prices for in-app purchases show up in US denominations ... and add up to around $150.
In any case, Theatrhythm is worth a try if you have even the slightest appreciation for Final Fantasy music. The 3DS version retails for $30.