OK, so the desktop connector provides the user with a connection between your local file and BIM 360 Docs. It installs a link a bit like installing Dropbox or Google drive locally so you can access the files directly from your machine without having to open the web application. Here is a link to the help online which explains this further.
Also when installed you will see you can access your files in Revit using the Recent Files (see below image). It allows you to access the hubs and projects you are a member of.

You won't have an icon within the open window because you then use the desktop connector. Not to say you couldn't put a shortcut on there.
The desktop connector looks like the below image. You can then select files which are in the folders within BIM 360 Docs. It also allows you to upload files directly from your desktop without opening the BIM 360 Docs application through the web browser.

You must install the desktop connector from here it only takes a few minutes to install. It does not come directly with the Revit installation. You will need desktop connector for both BIM 360 docs and design.
I hope the above makes some sense.
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Gareth Spencer
BIM Technical Consultant | Man and Machine (UK)