For geography maybe, but geodesy is a much more exact science

While computations of the great circle distance can be laborious, they are not especially difficult. Key things to consider are rounding of the angular coordinates (nearest arc-second vs nearest arc-minute), and the choice of reference Earth model (ellipsoid vs sphere).

If AA's great circle calculator uses coordinates rounded to arc-minutes, then results will differ from gcmap. Likewise, AA's calculator might simplify the math by representing Earth as a sphere, whereas in reality Earth is an oblate spheroid or ellipsoid (gcmap uses the WG84 ellipsoidal model as a reference).