Ryan already gave a very good answer. I just want to give a few clarifications on how one should think about PDEs in general. Most importantly: you should not think a differential operator as defined by "a formula". What do I mean by that? In your question, you wrote that the Laplacian in Euclidean coordinates is $\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$, but in Spherical coordinates it is not $\frac{\partial^2}{\partial r^2} + \frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial \phi^2}$. But why should one have thought that the two expressions are the same?
To illustrate, think of the function on the plane $f(x,y) = x^2 + y^2$. This represents the square of the distance of a point from the origin. In radial coordinates, the same function is written as $f(r,\theta) = r^2$. Not $r^2 + \theta^2$, which is a very different beast. Just like how the direct replacement $x \to r$ and $y\to \theta$ changes the function you are considering, the replacements $\frac{\partial}{\partial x} \to \frac{\partial}{\partial r}$ etc changes the operator you are considering.
Now: to think geometrically (on manifolds), a (real valued) function is an assignment of a number to each point on the manifold. Then in any coordinate system there is a representation of the function by a formula. Since we like to work with formulas, we can interpret the notion of a function as a "map". This map takes as its input a coordinate system, and outputs a formula that represents your function in that coordinate system. (If you are familiar with category theory, this is similar to how one can think in terms of arrows instead of dots.)
Similarly, a partial differential operator can be thought of as an object in itself. Our usual way of writing it as partial derivatives in some coordinate system with coefficients is just a convenient representation of the real object upstairs. Therefore, analogous to how a function is merely a "map" from coordinate system to the formulaic representation, you can think of a partial differential operator also as a "map" which takes as its input a coordinate system and outputs an expression which is a sum of partial derivatives with some coefficients.
The jet bundle formulation is just a sophisticated, rigorous way of formulating this idea. The main difficulty is making sure the intuition above is compatible with the change of variables formulas. Suppose we are given a coordinate system $A$, and an expression in terms of partial derivatives which we'll call $L_A$. The question becomes: does there exist some abstract partial differential operator $L$ such that its representation in the coordinate system $A$ is precisely $L_A$? And if there is such an operator, is the fact that, for some function $f$ (written in coordinate system $A$ as $f_A$), $L_A f_A = 0$ invariant under change of coordinate system? (That given any other coordinate system $B$, $L_Bf_B = 0$ also.) The tools of jet bundles allows you to make such a consistent definition/description of partial differential equations.