In general, you can never restore a backup from a newer version of iOS to an older version of iOS.
So - you can't load 6.1.2 backup on 6.1.1 or 6.1 or 6.0 or 5.x (etc…)
We're not a good place to ask specifically about iOS 7 until it's released to the general public, but this is known to not work for any OS categorically whether your backup is from a current, past or beta(future) iOS release.
You'll need to either run the OS needed for a particular backup or find an older backup or start over with syncing content to a freshly erased device.
Now, in iTunes open preferences and devices and look to see if you have more than one backup for that device. Also, when you restore from iOS - it shows you the latest backup from that device by default - but you can also tap a button to show other backups in case you have other devices or other backups that are older than the most recent backup. I hope you can find a backup here or perhaps restoring the backup folder on OS X by using Time Machine to restore an older version of the folder