The question was asked on Facebook, it was easier to follow-up and continue the conversation via email. Great, the customer was helped, everything’s good, isn’t it? Yes, but no. 
Did you go back and close the original conversation in some way to indicate where the conversation went?
While our focus is on the active conversation, it stays out there, meaning the next person to come along can read (and even try participating) but they won’t have the same understanding of the context you and your original participants did. It’s up to you to provide some clarity for those that will follow.
That extra step not only helps those that follow to understand, it also helps to manage perceptions. Just leaving the public portion of the conversation hanging gives a bad impression, even if you are really moving mountains in the background.