Office 365 is not actually an on-line version, it is installed on your computer using Click-to-Run technology and is significantly more capable than the Web Apps.
The applications in Office 365 are identical to those in Office 2013, the difference between the two being the method of licensing - Office 365 being subscription based that will always be kept up-to-date as long as the subscription is maintained while Office
2013 is a single, one time purchase licence.
If the Office 365 subscription is not maintained, the application will go into a reduced functionality mode.