Coordinates vs. Geometries: How can we know two coordinate systems describe the same geometry?
In general, you can't know whether two coordinate systems describe the same geometry.
This is a slightly surprising result, but it comes down to various computability results on the decidability of the equivalence of expressions: I think the key thing is Richardson's theorem which says that it is undecidable, in general, whether two expressions are equivalent, even if those expressions are only mildly complicated (certainly the kind of expressions that occur in practice are complicated enough). This is why, for instance, simplification in computer algebra systems is hard (and not possible in general).
So, in general, given two coordinate systems, with the metric expressed in each system, you can't know if they represent the same geometry. Or to be precise, there is no algorithm which will tell you this (perhaps a human could know by magic or something).
This has turned out to matter in the past. In particular, in GR, there is the problem of enumerating and classifying exact solutions to the field equations: given a solution (expressed as a coordinate system and a metric), how do you know if it is a really new solution, or whether it is an old one dressed up in new coordinates? Well, in general, you can't know.
But, of course, in very many cases, you can know, and there was a considerable effort in the 1980s to design tools which, given two solutions, would try to infer whether they were the same, sometimes with some manual intervention. I worked slightly on some of these tools.
Note, what I am not saying is that you can never know: very often you can: I am saying that you can't always know, because there's a halting problem, and further that this can matter.
A very good case when you can know is where you are given an explicit mapping from one coordinate system to the other, of course: you can check what happens to the form of the metric in that case, and check it's the same as the form in the new coordinate system. The problem arises when you don't know the mapping.
There is a Wikipedia page on exact solutions which probably has some good references. In particular I suspect that the Exact Solutions of Einstein's Field Equations (2nd edn.) by Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E is a good reference, although I do not own a copy.
Fortunately things are not so hopeless as @tfb describes. His answer is very general and could be given verbatim - almost -for any problem is any formalized discipline.
In this specific area an algorithm exists which can distinguish between 2 manifolds: Cartan–Karlhede algorithm.
In simple terms, if you want to compare two charts/atlases you check scalar invariants such as the Ricci scalar or the Kretschmann scalar. If any of these scalars are clearly different from one chart to the other, then the manifolds are clearly different.
If you use a CAS (Mathematica for example), compute the invariants for 2 different charts of the same metric and plot them for different ranges of coordinates. You will see that the plots are the same. Actually, inverting - at least locally- one invariant, you can find the functional relation between the 2 charts.
For more details check:
https://physics.stackexchange.com/a/167838
and also:
https://en.wikipedia.org/wiki/Cartan%E2%80%93Karlhede_algorithm
A coordinate change just amounts to a diffeomorphism. The geometry is unchanged by diffeomorphisms. Since the geometry is what the physics is, the physics is unchanged by diffeomorphisms as well. Different coordinate systems allow us to extract information in convenient ways, but they are all equivalent.