Copying a process should work ok. I wondered whether perhaps the original process had a template applied to it which set some values to send it down the right workflow? My only other idea is that you have window copy rules which are set on particular windows and which are published by a view rule which explicitly states the window should show for a particular process?
It might well be something simpler than this though but there's many things which could govern this. If the reason isn't clear, then I would suggest this is one to log with your support provider as they can compare the two processes to figure out what is going wrong.