You haven’t mention what windows you are using is it XP or Windows vista or windows 7. But I think your question is on XP. So I am giving answer on basis of windows XP. Microsoft don’t make product. It make product as service pack 1,service pack 2, and service pack 3.
So at first you have to know what service pack is your windows 2003 and then you have to know what service pack is your windows 2008. If it is same then you can use your old key on your new one, but if it is not then you have to buy a new key from Microsoft.