They are entirely different concepts. Your question is like asking "What is the difference between "temperature" and "weather"?" The term "damping" in physics typically refers to an effect that causes a reduction in the rate at which some process takes place. For the case of harmonic motion this can result in a reduction (over time) of the amplitude of a harmonic oscillation. "Friction" typically refers to a (dissipative) force acting on a system. Friction can be the cause of damping in a system.
So there are different kinds of damping on simple harmonic oscillators, but the most common type is a force linearly proportional to velocity acting in the direction opposite to the direction of motion of the oscillator. This works for a vertically oriented spring with an object hanging at the end, since that single damping force is drag, which is a type of friction for objects moving in fluids, but for that same unit oriented horizontally on a surface, there is additional kinetic friction between the surfaces opposing the motion of the object. That means there are multiple distinct 'damping forces', one drag and one of kinetic friction, and so the usual model of a damped oscillator no longer works. In other words, different kinds of friction can be forms of damping.