Testing for Riemannian isometry

Here's an irresponsible under-referenced response.

On the one hand, this is hard! Indeed, I've read somewhere --- not handy --- that this question lurks among quantum gravity's many difficulties. For instance, in general, given combinatorial descriptions of two topological manifolds, it's undecidable whether they are homeomorphic.

On the other hand, it's not too hard, in that given two metrics on contractible patches you can study how their curvature tensors act on orthonormal frames; and thus for a point in each you can ask if the curvatures are equivalent, which boils down to a not-too-large linear algebra problem (that much is essentially the answer to your first linked question) and if they are you can study whether such an equivalence extends to a neighborhood of those points, e.g. comparing the exponential maps (and the second linked question is a special case of this).

I suspect you could even do such a comparison in a Computable Analysis setting, assuming some kind of oracle access to the two metrics and their curvatures, to decide whether two metrics were Gromov-Hausdorf closer than $\varepsilon$, or Gromov-Hausdorf/Sobolev close as spaces-with-2nd-order-data --- but I digress.

This goes by the name of the "equivalence problem" in riemannian geometry and it is an important problem in the classification effort of solutions to gravity field equations. Malcolm MacCallum has many results in this area. For example this paper from 1985 might be a place to start reading about it.

It all depends on what you mean by "the same space". If you have two possibly different Riemannian metrics on the same underlying space (i.e. smooth manifold) and you want to know if the two metrics are the same (i.e., the distance between any two points in space given by each metric agree), then indeed all you need to do is check to see whether the symmetric matrices you get from local co-ordinates agree at each point in space.

If on the other hand, you have a metric on one space and another metric on a second space and you want to know whether there is an isometry between the two spaces (note that the two spaces could actually be the same space but you're looking for an isometry that is not necessarily the identity map), then problem is considerably more difficult (unless one of the spaces is special, like flat or constant curvature) and I defer to the answer by José.