It is well known that coordinates are just tools for representing the location of a point in space (generally speaking).
The same is to say a point $P$ is $(x, y) = (1, 1)$ in cartesian coordinates and $(r, \theta) = (\sqrt{2}, \frac{\pi}{4})$ in polar coordinates. We also know that representing specific things are much easier in some coordinates than in others. For example radially simetric systems, are much easier to express in Spherical/Polar coordinates.
My question is, while solving differential equations, why using an specific set of coordinates changes the result. I mean, coordinates are just a way of representing something, but that "something" is invariant between coordinates; a point is at the same location in spherical and in cartesian coordinates.
For example, imagine you are trying to solve the wave equation:
$$ \nabla^2u = \frac{1}{c^2}\frac{\partial ^2 u}{\partial t} $$
Depending on your coordinates you have one laplacian or other (which basically will determine the "shape" of your solution). For cartesian coordinates you have: $$ u(x, y, z, t) = A \sin(\omega t - \mathbf k \cdot \mathbf r + \varphi_0) $$ Just sines and cosines, but for spherical coordinates you get a monster similar to: $$ u(r, \theta, \phi, t) = R(r)T(t)Y_{lm}(\theta, \phi) $$ Where $Y_{lm}(\theta, \phi)$, are the spherical harmonics. This clearly is a much complete and intimidating equation, which basically arised from a change of variable. Is there an intuitive reasoning of why this happens? Why a change in the way you represent things, changes the solution of an equation? If coordinates are just different ways of expressing the same thing, would it be possible to get the cartesian solution from the spherical coordinates equation and vice versa?