I'm interested in these discrepancies, particularly as I've written flight planning software that makes extensive use of this calculation. It appears that Great Circle Mapper uses 3963 mi as the radius of the earth, which tends to be at the high end of the scale http://en.wikipedia.org/wiki/Earth_radius
Our software uses 3956 which is sort of midrange between the high and the low estimates and seems to correlate with what AC is using. The other source of difference seems to be exactly which latitude/longitude point is used in the calculation, e.g., control tower, runway, geographical center, etc. Since airports can cover many square miles this can result in several miles difference, depending on the points selected.