If one looks at the limit as light's mass approaches zero, Newtonian Physics predicts a deflection of light (this can be seen by the fact that all objects are accelerate the same due to gravity.) General Relativity predicts this based on curvature (making the mass irrelevant.) Why does General Relativity predict a significantly different amount of deflection, given that things like me are predicted to accelerate the same amount (or around the same amount) regardless of the theory used?
Let's say there's a straight railroad track in empty space. A large mass is brought near the track. General relativity says that the track will curve, while Newton's theory says the track does not curve.
Now let's say there's a fast train on that track, and the track and the train are free falling.
According to general relativity there's some deflection of the motion of the train caused by the curvature of the track. The faster the train moves along the track the faster it deflects.
When velocity of a train on the track approaches zero, the path of the train according to general relativity approaches the path of the train according to Newton's theory.
Both theories agree that in free fall the train does not exert a force on the track, so we may just as well remove the track.
Final conclusion: In curved space-time fast moving objects deflect fast.