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".

I very much prefer FF to IE, EXCEPT for the failure of FF to develop the same or a similar save link to desktop functionality that IE has had for years. IE saves the shortcut to the desktop WITH A FAVICON. Why can't the FF development team accomplish this?

I am using FF 4 and I was extremely disappointed that this basic functionality is STILL MISSING! COME ON, FIREFOX...WHY ARE YOU NOT AT LEAST MATCHING IE IN THIS? I KNOW YOU CAN DO IT IF YOU WANT TO...

IE is made by MS and thus has access to a lot of build-in OS features and that makes it possible to assign a favicon to an internet shortcut that opens the default browser.

This works:

-> go to Help Menu -> select "Restart with Add-ons Disabled"

Firefox will close then it will open up with just basic Firefox. Now do this:

-> Update Firefox to the latest version by going to Help Menu -> About Firefox -> click Check for Updates -> if an updated version is found then download the latest version and install it. After that follow these steps:

-> Update ALL your Firefox Plug-ins https://www.mozilla.com/en-US/plugincheck/

-> go to View Menu -> Toolbars -> unselect All Unwanted toolbars

-> go Tools Menu -> Clear Recent History -> Time range to clear: select EVERYTHING -> click Details (small arrow) button -> place Checkmarks on Cookies & Cache -> click "Clear Now"

-> go to Tools Menu -> Options -> Content -> place Checkmarks on:

1) Block Pop-up windows 2) Load images automatically 3) Enable JavaScript

-> go to Tools Menu -> Options -> Security -> place Checkmarks on:

1) Warn me when sites try to install add-ons 2) Block reported attack sites 3) Block reported web forgeries 4) Remember Passwords for sites

-> Click OK on Options window

-> click the Favicon (small drop down menu icon) on Firefox SearchBar (its position is on the Right side of the Address Bar) -> click "Manage Search Engines" -> select all Unwanted Search Engines and click Remove -> click OK

-> go to Tools Menu -> Add-ons -> Extensions section -> REMOVE All Unwanted/Suspicious Extensions (Add-ons) -> Restart Firefox

You can enable your Known & Trustworthy Add-ons later. Check and tell if its working.