The display is the control device that you use as an interface to communicate with the controller. the controller is the thing that regulates the motor's speed and power.. In a Bafang BBS01/2, the controller is attached to the motor and the control panel (display) is attached to the handlebars.
Both devices have sophisticated microprocessors in them, so they can do complicated things, and they work independently. They communicate with each other. The sensors (brakes, throttle, hall sensors, speed sensor and PAS are connected to the controller, so it makes some decisions for itself and can work on it's own. It also tells the control panel how it's doing regarding speed, and power and some other things. The control panel converts that data into the display so that you can understand what's happening.
Some manufacturers refer to the control panel as the human interface device (HID).