When I create a shortcut on the desktop for a web site site that I'm visiting
(Right Click/Create Shortcut), the icon for the web site is displayed intact
on the desktop.

But when I move this same shortcut to a desktop folder, the icon is replaced
by the classic "Big Blue E" Internet Explorer icon?

Why does this happen?

Any suggestions or fixing the problem, and retaining the original properties
of the Shortcut Icon when it is moved?

They were deleted when you emptied your Temporary Internet Files folder.

Read about it here...
[[Favicons are usually lost when you delete your Temporary Internet Files,
but FavOrg preserves them.]]
FavOrg
http://www.pcmag.com/article2/0,4149,108438,00.asp

Download it here...
108. Right hand side
FavOrg - Restore and Save Favorite Icons
http://www.kellys-korner-xp.com/xp_tweaks.htm

Direct download of favorg.zip
http://www.gratilog.net/anglais/internet/favorg.zip

These Icons are stored in >>
%%userprofile%%\Local Settings\Temporary Internet Files
or
C:\Documents and Settings\YourNameHere\Local Settings\Temporary Internet
Files

The contents of these files get deleted. Icons revert back to the generic
ugly IE e.