Google authenticator is absolutely horrible. Until recently there was just no way to back it up, at all. If you were getting rid of your phone, well, tough shit.
They now have cloud sync, which I don't really think is a good idea. But it solves your problem of migrating devices. However I've already moved on to Aegis, because I'm done fling scrolling through my Mahabharata of TOTPs to find the correct account.