As to the difference - all great circles are geodesics/geodetic lines, but only apply to perfect spheres. All geodesics are not great circles, since they apply to non-spherical surfaces as well.(i.e. ellipsoids)
The Earth is NOT a perfect sphere, therefore technically it has no Great Circles except for the equator (maybe - even that is now technically counted as just a "geodesic"). But for a very long time, "assuming" Great Circles was close enough. They got you there.
If I go any further, I'd just be quoting or rephrasing from this - so you may as well read the original itself:
https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid
Note the point that the Equator itself
may not be the shortest route connecting two places on the Equator (!!)
This may just be labels catching up to current geographical practice - is there a software difference between the 737s involved?
Or it may just be an option to follow tried-and-almost-true, known, quote-Great Circle-unquote routes, vs. more technically up-to-date (and shorter, fuel-saving) routes.