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Why does the Measurement tool display a different value for the same measured distance in a projected view versus an unprojected view?
In an unprojected view, the map units are unknown; thus, the Measurement Tool measures distances by calculating the great circle distance. This is a very accurate way to measure distances because the Earth is a sphere.
In a projected view, the Measurement Tool measures distances in the selected map units for the projection. The measurements are made in the uniform flat plane of the projection space. Because the Earth's surface is not uniform when distorted into a flat plane, the accuracy of a distance measurement varies depending on how the surface has been distorted by the projection. The Measurement Tool doesn't make any corrections for the distortion.
Due to the method used to calculate distances, the unprojected measurement is usually more accurate.