The Mute button is for the new message sound effect.
In Options -> UI Sounds -> Console Messages, you can enable sounds for new messages across several categories of channel. For example, If you have the sound enabled for the Global channels, then you will get a sound effect for any new message in Advertising, English Chat or World Chat. But if you want a more fine-grained control over the sounds, you can mute a specific channel using the right click context menu, while keeping sound effects for the other channels. However, the mute status of a channel doesn't seem to persist when you log off and log back on, unlike the menu settings.