Office 365 includes all the desktop versions of the Office programs; I am
guessing the Home edition you mention does not include the desktop version
of all seven Office programs as Office 365 does. The subscription means that
your Office programs will always be updated with new features as they are
added, whereas the standalone suites like the Home you mention are more
static to the time they were released and thus don't get updates other than
perhaps security fixes. With Office 365, you also get OneDrive space (1TB if
you buy the Personal subscription, 5TB for the Home subscription). With 365,
you can use the Office apps for iOS and Android. You also get a certain
number of Skype minutes (one hour per user per month if memory serves) for
calling landlines and mobile numbers. The Home subscription gives you the
ability to install Office on up to five computers, five tablets, and five
phones (the Personal subscription is one computer, one tablet, one phone).
The standalone suites are a perpetual license, whereas 365 requires you to
maintain a subscription for full functionality. Whether you go for a
standalone suite or 365, you will be using the same desktop programs, so
there is no difference with that in terms of accessibility. Of course, with
Office 365, you also have access to Office Online.

Regards,
Maria