The great circle route is the shortest route between two locations on a "roughly" spherical object like the earth. It is in effect the straight line between two locations along the surface of a sphere.
By the straight route - I assume you mean a straight line on a 2D map. This usually does not coincide with the great circle route. The difference in length between a so-called straight route and a great circle route would then vary based on both the map projection selected and the location of the two endpoints of the journey.