There are a few other parameters to consider. Launching out of Earth's gravity well is challenging. A 6.2-million-pound (2800-tonne) Saturn 5 rocket could put 260,000 lbs (120 tonnes) into low Earth orbit (4% of initial mass into orbit). Launching from the moon can put more payload into lunar orbit with the same mass of propellant. The lunar module weighed 10,300 lbs (4700 kg) including 5200 lbs (2400 kg) of fuel and could achieve lunar orbit with that small mass of fuel (~50% of initial mass into orbit, albeit with different fuels/efficiencies).
Once in orbit, because the Moon's mass is lower, its escape velocity is lower than Earth's as well and so is its Mars transfer orbit velocity. In addition, you can factor in the delta-V that you get from the moon's orbit around the Earth, which is 1 km/sec in the direction of a Mars traveler at one time during the month.
So from ground to ground (Earth surface to Moon surface vs. Moon surface to Mars surface) it is no surprise the Moon to Mars would require less propellant, especially considering you could aerobrake to slow the craft down at Mars. It would, however, take much longer. This also presumes you can manufacture a rocket on the moon. If you have to bring the rocket and its fuel from Earth, there's no point, except you might be able to slingshot past the moon after an Earth launch, using its orbital velocity to increase your craft's velocity as it approaches and passes, but because the Moon's mass is low, you don't get much of an effect.
From Clarke's perspective, it was true then and is true now. Propulsion techniques don't change the delta-V, only the time in which the delta-V is applied (assuming no massless propulsion) and the shape of the transfer orbit.