desktop icon for a site is a geneic icon not the icon displayed on the tool bar (it's a little picture)
The site identifier (at the extreme left in the URL of the address filed) of a Yahoo group is a little picture of three heads. When I drag the URL to my desktop, the little picture changes to a generic Firefox desktop icon -- the orange globe. If I bookmark the site, the icon on my toolbar is the little picture. But if I drag that icon to my desktop, it also changes to the generic Firefox icon. What must I do to get the icon on my desktop to appear as the little picture.
If you create an internet shortcut that way 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".
When I go through the same steps using Internet Explorer the icon retains its normal form. The difference between the type of shortcut created is:
Pinned Site Shortcut (.website) - Internet Explorer keeps icon
Internet Shortcut (.URL) - Firefox changes to generic