Damped harmonic oscillation differs from simple harmonic oscillation primarily due to the presence of a damping or frictional force, which causes the oscillation to eventually stop. In contrast, a simple harmonic oscillator is solely influenced by a restoring force. The discussion highlights that if an external periodic force is applied, it can create a driven, damped oscillator that continues to oscillate despite damping. The experiment mentioned involves using magnets on a glider atop an air track, where the magnets generate a damping force through induced currents in nearby conductors. This induced current creates a magnetic field that opposes the motion of the magnet, contributing to the damping effect.