Comment by thaumasiotes

Well, if a+b and a-b are difficult to factor, finding more congruences might give you another pair x+y, x-y, and if that pair is nontrivial but also difficult to factor, you may be able to make progress by looking at x+y / a+b and x+y / a-b. As long as they're not integer multiples of each other, the combination of the two facts is more informative than either fact alone, in the sense that you're now trying to factor a smaller number.