I'm genuinely curious as to what it is that Gmail (for example) doesn't provide yet that is so essential to dealing with emails. I gave a try to Mailbox but didn't see what the added value was, other than a couple UX gimmicks (some of which are now part of many email clients).
Off the top of my head - offline support, PGP support, multi-account support, decent mail-list/threading support, separation of client from service provider..
I'm actually quite a fan of the gmail interface; it is one of the most usable webmail experiences in my opinion (I use google apps both at work and for personal use). For day-long usage, I'd rather stick with mutt/vim.
It has labels, of which you can apply one or more of to a message. When represented in IMAP, it appears as folders (though you get duplicate mail objects for every additional folder that a mail is tagged with).
If you only apply only one label to a mail at a given time, it works as a folder would do just fine.
When Mailbox first came out, there was no Google Inbox, no snoozing, and the ios e-mail experience was lacking and sluggish in general. Mailbox fixed all that, and it was the first to properly do so that we trusted. We trusted it as it was by the well established Dropbox.
But since Inbox came out, I have seen no reason to open Mailbox, and I would say that Mailbox actually inspired Inbox.
Multiple accounts. I have multiple work accounts (several companies) and a personal one. Having each open in tabs/window would be too much. I use Apple's mail apps on Mac and iOS, with a single unified inbox.
Was there a killer feature that I missed?