Do you know what a 'favicon' is, do you care? It's a little icon 32 pixels square. So design wise it has to be simple, you can't get much detail into 32 by 32 little squares.
So where is this used? The most obvious place is at the top of the browser window. Look at the top now and you'll see a the green Cavespider roundel. If you have a number of tabs up you should see the little icon for each of the pages that should uniquely identify that website. It will also get used in listings of bookmarks, and many other places where you keep a reference to a website.
There is also the big brother of the Favicon, the apple touch icon. These come in a variety of sizes depending on the device used will effect which size is used. If you veiw a website on your smart phone and touch pad and you bookmark a page or it is listed in the history it will probably be presented by the icon.
Which would you prefer to see? Your icon on the screen or a default one?