The technical difference is how the overall thrust is steered:
* Quadcopter: the speed of four (fixed pitch[1]) propellers is managed independently. Spinning a single propeller a bit more tilts the copter away from that propeller[2]. This is very unstable, so speed needs to be adjusted at kHz rates[3][4].
* "Electric rocket": the thrust of the single propeller is steered using the control surfaces at the bottom to deflect the thrust. This is called thrust vectoring. Since the propeller is not directly used to steer, the speed doesn't have to be adjusted that often. However, the position of the control surfaces does.
* Helicopter: tilts the rotor plane[5] to tilt the aircraft, and inclines the individual rotors to change the magnitude of thrust. The rotor speed typically remains constant.[6] Helicopters are aerodynamically stable and don't require an automatic control loop [edit: no, see follow-up post!].
In theory the "electric rocket design" with a single propeller might allow higher efficiency: larger propellers are generally more efficient (thrust per watt). (However, the model in the video won't be very efficient, the props are quite small. And there'll be other tradeoffs too.)
Also, this is a refreshing take on RC aircraft that hasn't been done as much before.
Lift is generated on both sides of the propellers in both cases.[7]
Optional reading:
[1] Generally - variable pitch quadcopters exist. However that gives up the mechanical simplicity that's one of the main advantages of quadcopters over helicopters. Their advantage is that they can actively decelerate a lot more efficiently and also fly upside down easily.
[2] And turns the copter (yaw). Quadcopters use two pairs of counter-rotating propellers to be able to cancel the yaw.
[3] Quadcopters are unstable and need to measure linear and angular acceleration and correct the propeller speeds very quickly. Control frequency of 100Hz flies, 1kHz is somewhat standard, 32kHz is state of the art. (Note: Propellers have significant inertia. To give an idea, reversing prop turn direction in-flight takes ~1sec (and requires special symmetric props if it's supposed to work efficiently).
[4] This is feasible since the propeller is smaller (less mass, less inertia) than for helicopters.
[5] Due to gyroscopic effect, the tilt of the aircraft is 90 degrees out of phase.
[6] Assuming the typical variable-pitch helicopter. There are other options both for real life helis as well as for models (fixed pitch, coaxial, ...) but they're rare (or very cheap toys).
[7] Side remark: lift is generated due to flow turning, not due to different pressure differences caused by different airspeeds over top/bottom of wings as is taught in school. Search for "NASA incorrect lift theory" to learn more. https://www.grc.nasa.gov/www/k-12/airplane/wrong1.html