Restoring selected iOS apps and data to my new iPhone
Solution 1:
Update: apart from iCloud selective app backup which implies that you need to fully restore the new iphone, you can use this to back up and restore only selected apps and app preferences that go along.
Otherwise create another backup of your old iphone, this time via using iCloud. (Settings > iCloud > Storage & Backup > Manage Storage). From the options on the next screen, choose which and app data you want to back up exactly. Then hit "Back Up Now".
Upon restore to your new iphone only the app data you selected during the icloud backup will be restored.