Some of the reasons I still use paper:
Some counterarguments, if you don't mind! ;)
Off-line use. I can refer to paper copies and make notes on them even when I'm not around the computer.
This is actually a pretty good reason. Making notes on paper is pretty handy.
Audit trail. Most document-management systems and e-mail systems have document retention policies that're under someone else's control. Sometimes I need to control copies of the documents independently of company policies (eg. anything related to HR, records that might prove inconvenient for management later (like my detailing of exactly why something they want to do is a Bad Idea), etc.).
I might not fully understand you here, but I don't really see why you couldn't accomplish this without making paper copies.
Change control. Many times documents can be changed in the computer and, while it records that there was a change, there's no record anymore of what the document said before the change. The paper copies in my drawer can't be changed and I can pull them out to prove that yes that was what was originally specified.
What is wrong with just saving a copy someplace else?
Space. My desk's a lot bigger than the computer monitor, and I can lay out a lot more papers and diagrams on it than I can have visible on the monitor at one time. Very useful, that.
Get a big monitor. They're cheap these days. Use virtual desktops. That way you can handle much more than the desk ever could. And you can easily switch between cluttered virtual desktops, unlike real ones.
Reliability. I don't have to worry about the contents of my desk drawers and noteboard going *poof* when a system upgrade goes south and it turns out the restore process requires things IT can't afford to do.
You know, if my house went down with flames, I'd still have plenty of data around. It'd be all the physical things that I'd lose.