This is how I would define them.
The rate constant is the constant of proportionality between the rate of a reaction and the product of the concentrations of the rate-determining reactants. The higher the rate constant, the faster the reaction occurs.
The equilibrium constant is a measure of the extent to which a reaction will proceed towards completion at a certain temperature. Increasing the temperature will change the equilibrium constant depending on the dH for the reaction. An equilibrium constant >1 means the reaction will head towards completion. An equilibrium constant <1 means the reaction will tend towards not occuring. It is calculated by...