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According to Wikipedia:

The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles.

Note that the definition also mentions 'Geodesics'

Operationally what is the difference between them?

In ArcMap, the measurement tool allows measuring 'Geodesic' distances but not 'Great Circle' distances. Can these be considered equivalent?

A similar question regarding Planar vs Euclidean distance can be found at Difference between planar and Euclidean distance?

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Geodesics are the shortest path on a curved surface (e.g. sphere or ellipsoid) - like a straight line on a flat plane. A great circle is the shortest path on a sphere, but not on an ellipsoid (as long as both axes are not of equal length). So every great circle is a geodesic, but not every geodesic must be a great circle (on an ellipsoid for example)

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