Can you see the phone icon on the iTunes dash? I spent a while last night working out why iTunes wouldn't recognise my phone and once I got that fixed, I was able to get it backed up on my laptop (had to update the Apple USB drivers - plenty of guides online).
Argh this is driving crazy! I've updated everything, reinstalled everything, everytime I plug in phone, iTunes says welcome to your new phone, no option to back up just restore!
It sounds like the computer interpretes it as a "new" I phone as in new to that PC. It may be possible to get the stuff out of it using a thing called Copytrans was copy pod. Its the way I USED to be able to get my stuff off my I pod onto different computers.