Office and Office 365
Office 365 is the cloud (web) based version of Microsoft Office. You access it from your web browser (Internet Explorer, Chrome, Firefox) and must login using your school credentials.
Office 2013 is a suite of programs that resides on your machine. You can still access shared documents but you aren’t accessing it through your web browser.
Both sets of programs (Office 2013 and Office 365) are distinctly different but have a lot of overlap.