When a customer email comes in, we can rapidly discuss it internally using comments and callouts to each other, to come up with the appropriate response. This has worked so much better than endless email forwarding and replying within the organization before getting back to the customer. The discussion is linear, and in context with the external communications, so everyone is on the same page.
There are many ways to sort email into buckets, and sometimes too many:
Multiple inboxes, archived or open, assigned or not, replied-to or not, etc.
Then when you switch between the desktop app (OSX for us) and the mobile app (iOS), the default view of which buckets you see changes. So in the end, I do a lot of clicking to just see all mail, unfiltered, to make sure I don't miss anything.
Another problem is that it is hard to search for messages from a particular phone number. Dashes in the number will screw up the search.
We are able to send a huge quantity of outbound notifications via email and sms, using the API, tied to our other business applications.