The chat console is a web-based interface where you can receive incoming messages. Not all live chat tools offer such a backend -- some providers rely on an instant messenger like Google Talk (GTalk) or Pidgin.
Generally speaking, you have more options with a chat console. Everything can be accessed more easily; for example if you want to transfer, ban or pro-actively chat to your visitors. Tools like Zopim also give you the ability to use pre-defined text blocks (canned messages).
Click the image to see a screenshot of Zopim's dashboard.