A big BIG disadvantage is that there seems to be no way to go to a particular person's messages - there's no search that I can find, and you can't click on a person in a campaign to get to the thread with them.
This forces me to go to LinkedIn and manage them there
I'll add that the inbox has not been designed with UX/UI best practices seen with similar functions in other apps. It's clunky and renders slowly... not to mention not being real-time.