iPhone 5 won't turn on after restore

I have just upgraded from an iPhone 4 to an iPhone 5.

I backed up my iPhone 4, but when I try and restore the music/pictures/apps etc it doesn't work properly.

I plug in my new iPhone 5, and run through the 'Set Up Your iPhone' but restoring from my backup. It seems to complete the process within 4 minutes, and then says it will turn off and on again and will appear in iTunes. However it turns itself off, but then doesn't turn back on. I press the On/Off button on the top right of the phone and the Apple logo appears, and then goes to black again. The only way to turn my phone back on again is by doing a hard reboot (Home button + On/Off button at the same time). When it turns back it, it's as if its a new phone again - I need to select wifi, my iMessage details etc.

I then go back to iTunes and the restore doesn't seem to have worked, as the 'Set Up Your iPhone' page is back up again.

Weird thing is, I have all of my photos and have found a way to get my apps back - but I have no music, and iTunes still says my iPhone isn't set up.

I am really confused and not sure what to do.

Has anyone experienced something similar and been able to solve the issue?

Cheers