Comparing these distances:

${\displaystyle \pi R\sin \theta >2R\theta }$

for all $\theta$ except at the equator when $\theta =\pi /2$ when the distances are equal. This shows that the great circle distance is always shorter as of course the equator and the lines of longitude are all great circles.

A graph of the ratio of these distances shows that

${\displaystyle \frac{2R\theta }{\pi R\sin \theta }}$

tends to a limit as $\theta \to 0$. As we know $\frac{\theta }{\sin \theta }\to 1$ as $\theta \to 0$ we see that this limit is $2/\pi$ which is 0.6366 to 4 sig figs.