Quote: You should be able to take them between difficulties, most of them just can't be used to complete the related quest on those difficulties.
One thing to keep in mind is to be careful not to drop the quest item in a difficulty that you have already completed the quest in. If say, your character in hell joins a normal game for muling, and drops his hell diff Horadric staff/Kalims will.
No character who has completed act2 or act3 in normal may pick up the corresponding quest items. Also, no character can be holding the hell diff staff or flail and pick up the normal diff ones to complete the quests with. They must drop the hell version, then pick up the normal version to complete the quest. And once they have completed the quest, they can no longer pick up the hell diff staff/flail.
If your hell char drops his Hell Horadric Staff or Kalim's Will in normal/nm, kiss it goodbye. In multiplayer, there are several ways around this(as well as the "anti-rushing" locks), but in single player it can be *very* annoying.
EDIT: Basically the point I'm getting to is to mule off your Kalims Will or Horadric staff in act1/2 before you get to the point of completing the quests. I've before dropped a horadric staff in act3 by mistake, and then was *never* allowed to pick it up again(since no char past act2 can ever pick it up).