Restore from "iOS7 backup" to an iOS6 phone [duplicate]
You can edit the info.plist file. I successfully restored my backup to 6.1.4. Edit the build version to your ipsw's build version and product version to the version you want to downgrade to. Using plist editors in windows are highly recommended.
Windows: C:\Users[Your username]\AppData\Roaming\Apple Computer\MobileSync\Backup[Your UDID]
Mac: ~/Library/Application Support/MobileSync/Backup/[Your UDID]/
After that just restore the backup normally. If an error popped up at the end, ignore it. Hope this helps.
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 ~/Library/Application Support/MobileSync/Backup/