Updated May 2 2017. This article originally compared Office 2013 to Office 365. It applies to newer versions of Office as well.
Oh what a tangled web marketing can weave!
Using one name for both apps and services has got to be one of the worst decisions for consumers and support staffs. Users are confused and support is prematurely gray from trying to help users who say they have Office 365 and their email isn't working. It now takes a minimum of 3 questions: what type of email account, what version of Outlook, and is it Outlook on the desktop, Outlook in a web browser, or Exchange server?
There are two Office 365's: the applications and the hosted services. You can subscribe to either service separately or both together in one subscription. There are two Outlook's: the desktop app and the hosted service, formerly known as Hotmail with a logo that says "Outlook". Add in Office on Demand and Office Web Apps (which uses the same acronym as Outlook Web App) and it's easy to see why users are confused.
What were they thinking?!
Office 365, the applications, is (currently) Office 2016 with a few extra features added, like a larger SkyDrive. It does not include an email account, however, if you have a Microsoft Account (formerly known as LiveID or Passport) with a Hotmail, Live, MSN, or Outlook.com address you may be using that address for email. Office 2016 installed on Windows can connect to POP3 servers too. Or IMAP, or Exchange mailboxes (more than one!) You can even sync calendar and contacts with smartphones using Outlook or Office 365… that is, Hotmail's Outlook or hosted Exchange.
Office 365, the hosted services, gets the buyer Exchange server email, along with SharePoint, Lync and other services in higher priced plans. Some subscription plans even include a license for… Office 365 (currently Office 2016). BTW, anyone who needs to sync with mobile devices and owns their domains needs to consider Office 365 hosted services. I highly recommend the $8US E plan that includes SharePoint, even for small businesses or individuals (or the plan above it that includes Office desktop software). It comes with a public website and you can easily set up one mailbox to receive mail sent to multiple addresses, such as firstname.lastname@example.org and email@example.com or use public folders and shared mailboxes.
Office Web Apps is a lite online version of Word, OneNote, Excel and PowerPoint. It's available in OneDrive and can be used to read supported documents and create new documents. For a feature-by-feature comparision of desktop applications and Web apps, see Office Online Service Description.