I tried that and my phone is exactly the same but it doesn't sync the music I have put on the new PC. It just took off the music that I already had on there. It asks me if I want to transfer purchases and I click yes and it freezes for about 10 minutes then says it's finished and nothing is changed. My apps aren't listed in iTunes on my new PC. Nothing but stuff I manually added is in this iTunes, and none of of it is on the iPhone.
You might have to restore and intially sync it with the new computers itunes library.
I tried that too with a friends iphone and since it was synced with his computer nothing from my library would sync over to his iphone.
I copied my entire itunes library over to my new PC and was able to sync my iphone and ipad just fine without a problem. I think the most important file is the library.xml file (going off the top of my head - I could be wrong with its name) which contains a unique library identifier.
Is there a way I can create more than one backup? It only gives me one backup and I'd like to manually save the right one in case I mess up and it makes a backup of the new one that I didn't want and overrides the old one that I did want.