What is the difference between a direct drive hub motor and a geared hub motor on an e-bike?
The geared and gearless terms refer to the way power is transmitted from the motor to propel the bike forward. In practice, this distinction usually applies to hub motors (located in the center of the wheel) rather than mid-drives (located around the pedal crank) -- the vast majority of mid-drives are gear motors.
Geared hub motor overview
Geared hub motors use a gear system within the hub to transmit the motor's rotational force to the hub and wheel. It's easiest to see in a typical cheap hub motor in the picture above. A sun gear, fixed to a spinning motor, turns to the hub through a series of other gears (called planetary gears and ring gears, but that's not important...).
The main point to understand is that the gears inside the hub slow down the jerk of the motor, allowing the wheel to turn at a more appropriate speed. The motor likes to turn fast, much faster than the bike's wheels, hence the need for gearing. Geared hub motors are the most common and economical form of the hub motor.
Gearless (Direct Drive) Hub Motor Overview
Yes, you guessed it, gearless hub motors have no gears inside the hub! As you can see in the picture above, the hub in a gearless system is the motor itself. The bundle of copper wires you see is part of the stator on the right. When the motor controller pulls current from the battery into the wires, the stator becomes an electromagnet. In the hub shell on the left, are permanent magnets that force the stator to turn when energized by subtly attracting and repelling magnets at just the right time. There are no gears involved - only magnetism - hence the name "gearless".
While this sounds like the ideal solution - why introduce extra gearing in the hub if it doesn't have to be? - It's not as simple as we explain in the Pros and Cons section below.