The primary changes are:
1. Better connectivity with MS server products. You don't need that.
2. Uses xml internally. You don't need that.
3. New user interface.
After the 60-days trial you would have to decide anyway: buy or uninstall. If you are happy with 2003, I'd advice to install a second copy (if the license allows it, as you say) of it on your new laptop.
Switching to the full version on both computers isn't possible with the trial version, you'll need to buy the full version for that. But why should you do it now?
I bought a new computer (lap top) that has office 2007 trial version (60 days)(student) loaded but I have not opened the software yet. I have a office 2003 (student) that I currently use on another computer and still will be able to install on my new computer. My questions is there any real reason to change to the newer version of office, and if I buy the full version thats on the lap top will I be able to install on my other computer, and if so how would I do this as I only have the restore cd for the new computer but no office cd?