I don't think Mailbox does anything, really, you can't already do with any regular IMAP client, or just with Gmail but what it does do is make it very easy to quickly triage incoming mail. It's not magical, but it has a pretty solid interface.
What Dropbox probably means is better server infrastructure behind it, a faster rollout (hopefully) and probably some Dropbox sharing and attachment-based features down the road.
I already have a TON of documents in Dropbox already, so the paranoia factor is a moot point.