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Force constant is the constant of proportionality for spring force and displacement. 

Damping constant is the proportionality constant between damping/drag/friction force and velocity or velocity square.

Damping constant is small usually.  In rare media, it is neglected.  Force constant contributes to no loss in energy or momentum.  Damping constant slows down and dissipates energy.

Usually both constants are present when dealing with a real life spring oscillation.
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