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Laplace's equation is of the form Δ(x)=0, while the wave equation is of the form Δ(x)=(second partial wrt time)(x). Of course there are constants in the wave equation, too. The inhomogeneous form of Laplace's equation is Poisson's equation.
In more practical terms, the wave equation is exactly what it sounds like, describing wave motion/propagation. Laplace's equation is more abstract, describing, for instance, potentials.