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.
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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.
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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.