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".

If you're using Internet Explorer as your default browser, you can create a shortcut via drag & drop and it will take on the web site's favicon if there is one. FireFox just gives it the default icon for the browser.

It's certainly not a deal breaker (this is one of the VERY few things that's better on IE) but it would be helpful if FF's drag & drop shortcuts could also take on the site's favico automatically.

This add-on https://addons.mozilla.org/firefox/addon/9548 or cor-el's solution are ok for those whose shortcuts are more static in nature. However, I often drag a shortcut to the desktop for a site that I'll visit later on (because my bookmarks are a mess LOL), but will eventually delete the shortcut after visiting the site again, usually within several days. In my case, it's not worth the extra effort to manually save and assign the favicon for each shortcut, so it would be nice if this could happen automatically like it does with IE.