I had the same problem when I upgraded to a 5S from a 4. what you need to do is
1. After restoring your new iphone using your old iPhone iCloud backup, back up your new iPhone with iTunes.
2. Go to your new iPhone's settings - General - Reset - Erase All Content and Settings
3 Once your phone resets completly go through regular activation and AGAIN (this is important) restore using your old's phone iCloud backup AGAIN!!!!.
4. After your new iPhone is back on BUT, BEFORE backup completes go to your new Iphone's settings - iCloud - Storage & Backup - Scroll down and at the bottom tap on STOP RESTORE!
5. After thats done go to iCloud - Storage and backup - Manage Storage - tap on your old iphone's back up and delete! VOILA!
6th and Final step. go to itunes with your new iPhone pluged in, click on restore backup and choose the most recent backup and DONE!
NOW YOU CAN FREELY CHANGE YOUR BACKUPS TO ICLOUDS INSTEAD OF ITUNES!