In both cases the answer is that standardized tests have roughly as much predictive power as GPA (SAT is about as powerful as HS GPA; GRE similar to undergraduate GPA). Not bad for a brief test! When these factors are combined the overall predictive power is increased. My opinion is that standardized tests load more heavily on cognitive ability and less on conscientiousness relative to course grades, hence the non-redundant information in the two measures.

This is a meta-analysis of GRE validity by Kuncel and collaborators. Here are slides from a related talk. See also this blog discussion (social scientists) of the GRE.
I first came across Kuncel's work when I was investigating SAT validity using U Oregon undergraduate data. Unlike many of the people working on this topic, Kuncel is neither an idiot nor an ideologically motivated faker. For example, he understands issues like restriction of range, variation in course and subject difficulty, and self-selection effects. See these slides from his talk on SAT validity. Video and audio here (bonus: Thomas Espenshade; Kuncel starts at 27 minutes in).

Validities for various standardized tests:
