Home Assistant is a Python program, in simple words. It can be run on various operating systems and provide the ability to track, control and automate your devices. When people talking about Home Assistant they usually refer to a standalone installation method.
Hass.io is a combination of Home Assistant and tools which allows one to run it easily on a Raspberry Pi and other platforms without setting up an operating system first. Hass.io is an all-in one-solution and has a management user interface that can be used from the Home Assistant frontend. This interface is not present in a standalone setup of Home Assistant.
Be aware that add-ons are only available in Hass.io, due to the way Hass.io is installed.