How much shorter is the great circle route then the straight route?

The great circle route is the shortest route.

It is just that if you plot a great circle path on a flat map (unless it is the equator or a line of longitude) it will look bent - this is an artifact of map projection.

The actual difference in route lengths depends on the routes you are comparing.
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.

The SHORTEST line between two points on a sphere IS a GREAT CIRCLE.

Also, by definition, the shortest line between two points IS the straight route.