Hey Guy,
Yes, this is a big pain and part of Apple's unfortunate "iTunes controls everything" approach to the iPod/iPhone family.
There are some workarounds.
1) Apps: The good news is that Apps are remembered in the iTunes store and you CAN move apps off the phone onto a new computer. Connect the phone, open iTunes, and do the following to move your iPhone Apps off the phone onto the computer: a) Right-click (control click on a Mac laptop) on the iPhone icon; b) Select "Transfer Purchases"; c) authorize the computer if necessary; d) wait. iTunes will then pull the Apps off your iPhone, put them in the "Mobile Applications" folder inside the iTunes directory, and authorize them. You are now good to sync and go.
2) Contact & Calendar: The best solution here is to cloud-sync the data to another service so that you can get it onto the computer. Just fire up a MobileMe (ick) or Google GMail account. Configure your phone to sync to the cloud service, and you should be able to "put" all the data onto their server. Then you can sync it "down" to your computer. Undo the cloud sync and revert to USB sync and you are good to go. Although, maybe you just keep the cloud sync anyway since it's better. Full disclosure: I think MobileMe has a lot of problems. My experiences with it have not been good.
3) Music & Photos: These are tricky. I think the best bet is to use a third-party greyware application. You are not really supposed to be able to access music & photo files on an iPhone except through iTune and iPhoto. Of course, the wonderfully smart people out there in the hacker community don't take that one laying down. Here are two such solutions. I have not tried these myself, but Senuti is well-established and has a huge following. The ecamm product looks really interesting; I've not seen a solution yet for accessing SMS threads and Notes. If it works, it's worth the money.
http://www.versiontracker.com/dyn/moreinfo/macosx/23390
http://www.ecamm.com/mac/phoneview/
Good Luck!
Hopefully Apple will loosen the reigns in the future -- I'm betting that this issue of non-computer-backed-up content is even worse on the iPad . . .
Happy Restoration!
Message was edited by: r_marshall