No, you can use the copy you have (if it is a full version). Most vendor versions (Dell, HP, Compaq) are not full versions, and won't work on a different system.
Since you are reinstalling it, and not just putting it on an additional computer, you are in the clear legal-wise. Actually, you could use a different Windows CD as long as you use the code that came with the Windows on your current system (if you don't have a full version).
Windows takes a snapshot of your system. When you boot, it checks what it finds against what it knows. If the two doesn't match up, Windows won't boot. Changing mobos is a major change, and Windows will see it.