Microsoft is listed as the #35 company in the world for 2013, according to the Fortune 500, but sometimes their product branding can be unclear. There are two different services that they offer and both are called Office 365, but the two are almost completely different from a feature set. In this article, I will describe each and what the appropriate uses are for them.

For Home: the full name is actually Office 365 Home Premium and it basically boils down to a "leased" version of the Office software suite plus a few online extras. The Home subscription costs $99/year, or if you prefer not to commit as long, you can get it for $9.99/month. This entitles the buyer to the Full Office Professional Plus software for up to 5 computers in the home. This includes Outlook, Word, Excel, PowerPoint, OneNote, Publisher, and Access. To "purchase" a single copy of this suite would cost $399 for one computer, so the combined value of this is $1995. At $99/year, it would take over 20 years to get to that cost. Additionally, the subscription entitles you to the newest versions of Office whenever they are released.

Along with the Office software, you also get online "web app" versions of the programs and 20gb of online storage space (SkyDrive) for saving and sharing documents with the whole family.
For Business: This is also a subscription service, but in its most basic form, its focus is on online email boxes for small business. The pricing is $5/mailbox/month when purchased yearly ($6/month when paid monthly). This enables access to a 25gb Exchange email box, calendar, contacts, and tasks, all available through Outlook, a very capable webmail client, or mobile devices (iphone/ipad, Android, and Windows phone support it out of the box), and all are always in sync with one another (i.e. when you file something in one place, it's filed in all; delete something on one device, it's deleted on all of them).

Also included is SkyDrive Pro (also known as SharePoint online), allowing each licensed user to save 7gb of information of their own and also 10gb (+ 500mb per user) for shared space. There is also an instant messaging system included called Lync, which allows business users to send quick text messages, share desktop applications for collaboration, and communicate via audio and video.
For a premium price, you can add the Office 2013 Professional Plus Suite as is the primary inclusion in the Home version above.
It can be confusing to understand the full matrix of features available in each of these options, but each is a great option for their intended audience.