Are there any modern browsers that won't detect the favicon.ico automatically? Is there any reason to add the link tag for favicon.ico?
<link rel="shortcut icon" href="/favicon.ico">
My guess is that it's only necessary to include it in the HTML document if you decide to go with GIF or PNG...
To choose a different location or file type (e.g. PNG or SVG) for the favicon:
One reason can be that you want to have the icon in a specific location, perhaps in the images folder or something alike. For example:
<link rel="icon" href="_/img/favicon.png">
This diferent location may even be a CDN, just like SO seems to do with <link rel="shortcut icon" href="http://cdn.sstatic.net/stackoverflow/img/favicon.ico">.
To learn more about using other file types like PNG check out this question.
For cache busting purposes:
Add a query string to the path for cache-busting purposes:
<link rel="icon" href="/favicon.ico?v=1.1">
Favicons are very heavily cached and this a great way to ensure a refresh.
Footnote about default location:
As far as the first bit of the question: all modern browsers would detect a favicon at the default location, so that's not a reason to use a link for it.
Footnote about rel="icon":
As indicated by @Semanino's answer, using rel="shortcut icon" is an old technique which was required by older versions of Internet Explorer, but in most cases can be replaced by the more correct rel="icon" instruction. The article @Semanino based this on properly links to the appropriate spec which shows a rel value of shortcut isn't a valid option.