The importance of the Laplacian is a reflection of the importance of Riemannian geometry, both for its own sake and in these other fields. (Obviously, absent a generalization, you can't have a Laplacian without a Riemannian metric.) In Riemannian geometry, the Laplacian is the first scalar linear differential operator available which is "covariant", i.e., that depends only on the Riemannian structure and not on extra choices such as coordinates.

It's natural for the first non-trivial possible structure of a given type to be fundamental. One reason is that it can be an approximation to something else with higher-order terms. For example, in the wave equation, if it is meant as a realistic model of sound waves, actually there are all kinds of higher order, non-linear effects; but one begins with the Laplace operator as the correct approximation for small waves. It has to be correct because it's the only one available. The same thing happens with heat and the heat equation.

Actually, something interesting happens if you relax the use of the word "scalar". If you have a spin manifold, then there is a Dirac operator, which is a Laplacian-like operator that is just as important. But even apart from that construction, there are various other operators, such as the Hodge Laplacian, that act on vector and tensor fields rather than on scalar functions. These are not exactly the same as the original Laplacian, but they tend to get the same name.