I'm not familiar with physics jargons compare to those who study physics professionally, but to remark my knowledge regarding this question, the Klein-Gordon equation is a relativisitc equation for particles (or matter waves) derived from the relativistic energy equation (here the units are natural) $$E^2=p^2+m^2$$ by looking $E$ and $p$ as operators acting on complex functions (which are in a Hilbert space) as follows $$\hat E\doteq i\frac{\partial}{\partial t}\qquad\text{and}\qquad \hat p\doteq -i\nabla$$ thus we get $$\left(\square+m^2\right)\psi=0.$$ My question is, I don't know how this equation can be applied to tackle real world problems, such as the wave functions of electrons in a hydrogen atom, with the absence of potential term (thus I can only think of the free particles case). To compare this with the Schrödinger equation, what you need to do is identify the coulomb potential field and put it in the equation then solve, so you can say the solutions are pretty much potential dependant, which I think is quite straightforward to understand, at least the idea of what-we-are-doing itself.
Remark. (I somewhat found an answer to this question, but as the question has been closed, I leave the answer here.) The Klein-Gordon equation is for free particles/fields. One needs to add interaction terms to introduce interaction with other fields. As an example, for the hydrogen atom, the photon field can be introduced to do the physics (per quantum electrodynamics, quantum field theory).
