I think the level of confusion exhibited by several commentators here only service to illustrate the complete Horlicks Microsoft has made of its product branding given that "Office 365" is an umbrella term for a series of different and often overlapping products aimed at both consumer and enterprise markets and which provide the end user with differing functionality depending on the package they have purchased. To whit:
Office 365 is the name of the cheap as chips cloud-based productivity suite of Word Processing, Spreadsheet and Presentation apps running in a browser and which share a user interface and most functionality with the legacy desktop Microsoft Office product.
Office 365 is also the name of the subscription service which allows the home user to install desktop apps on up to five devices, These apps are fundamentally Microsoft Office but benefit from a regular stream of feature enhancements and updates not immediately available to purchasers of the offline retail version. Purchase ALSO entitles the user to access the cloud based versions of the apps, and if we've been good users and configured our desktops to save to OneDrive then files can be accessed via the cloud on any device. Unless your desktop sync has glitched with an unhelpful error message.
Office 365 is also the name of the suite of enterprise cloud services which, depending on purchased tier, allow businesses to run filestore, mail and even AD systems in the cloud without recourse to any on-premises hardware. Billed on a per-seat basis it also allows use of the cloud Office apps and installation of desktop apps, unless you are on the cheapest tier in which case you are cloud only and it is no install for you bad customer.
Clear as mud isn't it? Don't get me started on Skype for Business which is the offering previously (or in certain backwaters of the admin console still is) known as Lync and an entirely different service to the product also called Skype as marketed to the consumer space. Or the fact that higher tier Office 365 (enterprise) customers get to use Sharepoint services for enterprise cloud storage, despite also having access to the enterprise-grade tier of OneDrive which performs largely the same functions on a basic level, just not in a granular Sharepoint-y way.
All of the above may be wrong as they may have changed stuff since last week.