They don't work with other people's IMAP/POP3 servers?
(Well, the Gmail iOS app certainly doesn't; I haven't tried the Outlook client but the description in the app store suggests a very limited set of providers.)
Outlook (I assume you're referring to the mobile app, not the desktop client) puts all of your emails on MS's servers, so that's a no go for lots of people.
Last I checked, the gmail app only worked with gmail accounts.
The "delay for later" feature is tremendously useful compared to traditional inbox handling.
Of course, to trust that feature, it also has to be tremendously reliable, and that's the biggest thing that Mailbox got right in the first place. That's also why it very handily beat out Mail Pilot, which had the same idea and got started earlier but to this day is still filled with terrible bugs.
I'm not the parent but Outlook's IMAP support is quirky at best and I very strongly prefer native applications to web applications. I also host my own email which marks gmail right out.
Email should be simple but there has been essentially no lasting innovation in email clients in a very long time unless you really want Google to host it for you
Gmail on mobile still doesn't have a unified inbox, and it also doesn't have features I've come to depend on such as snoozing messages. I usually use Inbox by Gmail on mobile, which also lacks a unified inbox and (bizarrely enough) support for HTML signatures but does have pretty good message snoozing, and I like how it handles bundling.
Outlook (incidentally, another example of my original complaint, as it's just former email startup Acompli with a fresh skin although obviously it's probably not getting shut down anytime too soon) doesn't have very good message searching for Gmail, which is a common complaint I have for non-Google mail apps. Obviously, I get that Google has a huge built-in advantage here; but if your email app is going to support the "Mailbox paradigm" of abandoning folders/traditional labels in favor of just marking things as done, it also needs to have search functionality that can back it up. Outlook was also really buggy for me when I last tried it, although that was a little while ago so it may have improved.
The new Outlook app is actually surprisingly good. I think the Sunrise guys (sunrise.am) are running that team and it shows. I laughed the first 2 times friends recommended me the app, but I tried it and now also highly recommend it.
My favorite tiny detail is that I can tune the notifications to push only unread counts to the icon badge and only for the "focused" inbox. That means I can avoid constant push notices for email yet still be able to glance down when I want to and see if I have any unread emails that are actual human communications.
Message searching for Gmail in Outlook iOS/Android is piped directly from Gmail itself. Your search is forwarded on to them and results are displayed back in the app. Immediate search results are from on-device and then filled with the Gmail results after they are received. So if the results are bad, then maybe Gmail has a secret API they use to make their app better.
Just curious, how is processing hundreds of emails in Mailbox or Inbox better than Outlook? Seems like trying to do that on a mobile device would be hard no matter what app.