Do you have to include <link rel="icon" href="favicon.ico" type="image/x-icon" />?

If you don't call the favicon, favicon.ico, you can use that tag to specify the actual path (incase you have it in an images/ directory). The browser/webpage looks for favicon.ico in the root directory by default.

You should in fact do both, so that all browsers will find the icon.

Naming the file "favicon.ico" and putting it in the root of your website is the method "discouraged" by W3C:

Method 2 (Discouraged): Putting the favicon at a predefined URI
A second method for specifying a favicon relies on using a predefined URI to identify the image: "/favicon", which is relative to the server root. This method works because some browsers have been programmed to look for favicons using that URI.
W3C - How to add a favicon to your site

So, to cover all situations, I always do that in addition to the recommended method of adding a "rel" attribute and pointing it to the same .ico file.