Though I'm not necessarily a proponent of automation or integration of sending emails (getting ridiculous amounts of auto-generated/useless email is pretty prevalent at many companies, and I certainly don't like it, nor think it is effective), I think tracking the flow of conversation is very important.
Especially when it comes to customers, keeping in mind everything the customer has communicated and having it at arms reach is very important, and also probably fosters some feeling that someone is looking at their case personally, and has been paying close attention to the back and forth (even when you haven't).
