I do not understand "does not have access to the Word Count facility"; is it not there on the Tools menu, or is it grayed-out, or does it simply not do anything?
It simply does not appear on the Tools menu (although I can use the Word Count toolbar).
It's the only difference between the two administrator accounts, as far as I can see.
Sounds to me someone has personalized menus on. You should see the double arrow at the bottom of the menu double click it and the entire menu with all of the options should appear. Another thing is the menus and toolbars have been customized. If that is the case you'll have to right click on a toolbar and select customize.