Home Assistant Core 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 Core they usually refer to a standalone installation method.
Home Assistant is a combination of Home Assistant Core and tools which allows one to run it easily on a Raspberry Pi and other platforms without setting up an operating system first. Home Assistant 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 Home Assistant Core setup.
Be aware that add-ons are only available in regular Home Assistant installations.