Microsoft's Office for the iPad only opens and saves files locally, or to OneDrive, OneDrive for Business, and SharePoint cloud storage. (There is no automatic syncing; you manually have to manage files for offline access.) There's also no support for Open In in iOS, so you can't send files to other applications or storage services in iOS. In iOS and Windows, you can open and save files locally, as well as from and to virtual drives for storage services.
In Windows, the default storage location is OneDrive, OneDrive for Business, and/or SharePoint, based on the user's account. It's extra work to save to local drives and storage services' virtual drives. In OS X, Microsoft's OneDrive and SharePoint storage services -- but not OneDrive for Business -- are accessible through an awkward add-on tool called Microsoft Document Connect that makes using Microsoft's cloud storage services a real pain; the non-Business version of OneDrive can also be accessed as a virtual hard drive.
As you can see, Apple and Google both strongly discourage the use of third-party cloud storage systems in iOS (and, in the case of Google, Android). Microsoft essentially forbids it and favors its own cloud storage services in Windows, as Google does for its Web-based Google Drive editor. Apple is the most open to other storage systems, but iCloud is its default.
At one time, Box, Dropbox, and their ilk promised a universal storage system for any operating system that mattered: Windows, OS X, iOS, and Android. That meant you could work on the same documents on any of several devices and applications, as well as make them available to other users. They were the glue for the fractured mobile world, not to mention between mobile and desktop devices.
No more -- you have to work hard to use Box, Dropbox, and their ilk from iOS productivity apps. For all practical purposes, iOS is the mobile business platform, so cutting out third-party storage services from iOS productivity apps essentially eliminates their ability to be a centrally managed, universally accessible conduit for business documents.
Why even bother with Box, Dropbox, and so on?
Enterprise hopes for managed cloud storage are dashed
A couple years ago, Box CEO Aaron Levie anticipated that Apple, Google, and Microsoft would provide free cloud storage as part of their ecosystems, making services like his commercially unviable for consumers. His answer: Focus on enterprises that want to manage documents in user-accessible environments -- and pay for the privilege.
But look at what's happened: The major tool in most businesses is Microsoft Office, and Apple, Google, and Microsoft have made access to Office files free but captive to their ecosystems. In other words, Box and the rest have targeted the enterprise space that Apple, Google, and Microsoft are making hostile to third-party storage service providers.
It's true that Apple, Google, and Microsoft have no enterprise management tools for their mobile clients -- you have access, or you don't. But so what? For the documents that really count -- Office documents -- you can't use Box, Dropbox, ShareFile, or other services instead, even though they provide such management capabilities. Poof! There goes the core value proposition of document management, at least in a core area. (I have no doubt Microsoft will at some point extend its Information Rights Management and Intune/System Center technologies to OneDrive and to mobile SharePoint access, but that could take years, so IT will be in an awkward position.)
In the meantime, if you want to use Office documents on mobile devices, you go with the cloud platform that comes with the mobile app you've chosen. This likely means businesses will face having documents stored in two or more cloud storage services based on employee app preferences. Whatever is stored in Box, Dropbox, ShareFile, and so on will be essentially orphaned, requiring real hoops for users to jump through to access -- and especially to save back to.