Move All data from old iPhone to new?

I have an iPhone 5 on iOS 7.1, and I want to move everything onto my new iPhone 5s on iOS 7.0.4

Is there any way to do this without *updating*?

I mostly just want to transfer over the messages/contacts/ photos/app game data, notes

I made a full backup but it does not seem to add anything to the 5s when I restore. I believe I saw a message once saying the backup may not work due to using an older iOS

Nope