Is there necessarily a difference? Some fundamental problems, including such simple things as the physical 3-body problem, don't have analytical solutions. That doesn't mean that approximate solutions solve an approximate problem. Algorithms can be designed to give an approximation to the optimal solution of the problem, which is known to be achievable only through ultimately numerical methods.
I don't know whether that is the case here: it is possible that they solve a different problem from the fundamental problem. However, even in that case: if the transformations made to turn the fundamental problem into a more tractable problem are generally applicable, then is there really a difference between solving the fundamental problem and solving the transformed problem?
I don't know whether that is the case here: it is possible that they solve a different problem from the fundamental problem. However, even in that case: if the transformations made to turn the fundamental problem into a more tractable problem are generally applicable, then is there really a difference between solving the fundamental problem and solving the transformed problem?