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As much as I understand it, a geodesic line of curvature is a line on the surface such that the projection on the tangent plane of its curvature vector is $0$ at every point. Principal lines of curvature are lines such that their tangent direction is coincident with one of the principal direction on the surface at every point.

But I can't see the difference between them. Can anyone make it clear for me?

Antoni Parellada
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Mykolas
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    One obvious difference: through a given point there are geodesics going in any direction, not only in the principal directions. – Hans Lundmark Jun 20 '12 at 14:08
  • wow, that was clear. Thanks a lot. Stupid me – Mykolas Jun 20 '12 at 14:20
  • @HansLundmark I guess it can be extrapolated that all principal curvature lines are geodesics; yet, not all geodesics are principal curvature lines? – Antoni Parellada Apr 22 '20 at 16:36
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    @Antoni Parellada No, consider any curve on a sphere that is not a great circle. Since a sphere is umbilical, every tangent vector at any point is a principal direction, so every curve on a sphere is automatically a principal line of curvature. However, only a great circle is a geodesic. – Ernie060 Apr 23 '20 at 20:44
  • @Ernie060 That is a perfect counterexample! Thank you! – Antoni Parellada Apr 23 '20 at 20:58

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