How exactly are conserved quantities, i.e. mass, energy and momenta of a system computed in special relativity and why doesn't it work in general relativity? I know that the curvature of the spacetime seems to be the problem in general relativity, but can't completely pin it down. Is it correct to say that in GR because the information about matter and gravity is encoded in the curvature of the spacetime, it's impossible to get any information locally (due to lack of invariance under coordinate transformations), so that integrating e.g. the stress-energy tensor is useless?
In special relativity, mass, energy, and momentum are unified in the momentum four-vector, so let's just agree to call this momentum.
Because momentum is a vector, not a scalar, a momentum at point A in spacetime lives in a different tangent space than a momentum at some other point B. That means that there is no unambiguously correct way of adding these two momenta. In order to do that, you'd first have to parallel-transport one of them to where the other is, but parallel transport is path-dependent.
Another way of putting this is to ask, if you could express the total momentum of a region of spacetime as a vector, then in what frame of reference would that vector be expressed?
In an asymptotically flat spacetime there is an answer to this question, which is the frame of a distant observer, and indeed in such a spacetime we do have a conserved momentum vector.
We do have local conservation of momentum in GR. This is expressed by the divergence-free property of the stress-energy.
Energy, linear and angular momentum conservation laws in flat spacetime follow from time, space and rotational symmetries of Minkowski spacetime. General curved spacetimes have no such global geometric symmetries.
Other symmetries could come from gauge symmetries or symmetries of the lagrangian of a theory defined over a curved spacetime. The physical intuition of why the possible symmetries that may arise from the idea of the last sentence is that gravitational collapse is in principle possible for a wide class of solutions to Einstein gravity equations. That is relevant because the Penrose-Hawking singularity theorems imply that black holes are the endpoint of a general collapse and the number of conserved charges in black hole backgrounds is strongly constrained by the well known no-hair theorems.