Suppose I'm chatting with Pitter.
What is the difference between "Mute conversation" and "Turn off chat for Pitter" on Facebook?
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"Turn off chat for Pitter" means that the chat will not show up in any of your chat interfaces, unless you explicitly open it in one - the conversation will use an "email" like interface where available instead of a "chat like" one.
"Mute conversation" means that you will not be notified of any new messages in it through any interface