If you drag a link or favicon on the left end of the location bar onto the desktop to create an internet shortcut then that shortcut gets the icon of the default browser.
If you want a different icon (favicon) then you have to assign that icon yourself to the desktop shortcut (right-click: Properties).
You can usually get the favicon if you append "favicon.ico" to the main domain of a website (e.g. http://www.mozilla.com/favicon.ico ) to display the favicon in a tab and save that image to a folder or see if you can find it in "Tools > Page Info > Media".

OK, that works, but it sure is a lot more complicated than the point-click-drag that works in IE. Oh well, maybe in a future version of Firefox they can incorporate this ability. Thanks.