If one looks at null geodesics (the geodesics of light) in the Schwarzschild space-time, I believe one will find that there is more deflection of the former than the later, so they definitely won't be the same.
Consider a light beam moving in the ##\hat{\theta}## direction at the point of closest approach to the central mass in an orthonormal basis of the Schwarzschild metric with coordinates (t,r,##\theta##,##\phi##). I've followed MTW's notation by using 'hats' to indicate the basis vectors of the orthonormal basis, to distinguish them from the coordinate basis.
Compare it to a purely spatial geodesic "moving" in the ##\hat{\theta}## direction. We have some affine parameter s for both the null geodesic and the spatial geodesic, the null geodesic has components ##dt/ds## and ##d\theta/ds##, while the spatial geodesic only has a component ##d\theta/ds##.
Both geodesics will be deflected in the ##\hat{r}## direction, but the magnitude of the deflection will differ. As mentioned previously using slightly different notation, the spatial geodesic will have a 4-velocity in the orthonormal basis of (0,0,1,0), while the light beam will have a 4-velocity of (1,0,1,0).
We will assign the indices 0,1,2,3 to the t,r,theta,phi coordinates, respectively. The nonzero 4-velocity components are all unity, and we will assume the separation vector is unity as well, to make the math easier.
Then the geodesic deviation of the spatial geodesic will have one nonzero component, ##R^{\hat{1}}{}_{\hat{2}\hat{1}\hat{2}}##, while the geodesic deviation of the light will have two nonzero components, ##R^{\hat{1}}{}_{\hat{0}\hat{1}\hat{0}} + R^{\hat{1}}{}_{\hat{2}\hat{1}\hat{2}}##
We see the light has a higher geodesic deviation. I actually expected the light deflection to be double, but that's not quite what I'm getting at the moment, I'm not positive whether my calculation or my expectation was wrong. But one can definitely see the geodesic deviation between nearby null geodesics is greater than the geodesic deviation between nearby spatial geodesics, so they aren't the same. In words, "gravity" and "spatial curvature" both deflect the light, while only "spatial curvature" deflects the spatial geodesic.
One can also see that the light beams do diverge, as do the null geodesics, consistent with the negative curvature. But they diverge at different rates.