When you're serving an .ico file to be used as a favicon, it doesn't matter. All major browsers recognize both mime types correctly. So you could put:
<!-- IE -->
<link rel="shortcut icon" type="image/x-icon" href="favicon.ico" />
<!-- other browsers -->
<link rel="icon" type="image/x-icon" href="favicon.ico" />
or the same with image/vnd.microsoft.icon, and it will work with all browsers.
Note: There is no IANA specification for the MIME-type image/x-icon, so it does appear that it is a little more unofficial than image/vnd.microsoft.icon.
The only case in which there is a difference is if you were trying to use an .ico file in an <img> tag (which is pretty unusual).
Based on previous testing, some browsers would only display .ico files as images when they were served with the MIME-type image/x-icon. More recent tests show: Chromium, Firefox and Edge are fine with both content types, IE11 is not. If you can, just avoid using ico files as images, use png.