Since I actually work for the Smithsonian Institution, I can assure you that that is the only form that is correct. "Smithsonian Institute" is simply a misinformed usage.
However, I do work for a research branch of the Smithsonian that is called the "Smithsonian Tropical Research Institute." AFAIK, this is the only branch of the Smithsonian that is properly called an Institute (but this applies only to this particular branch). We have a bunch of Museums, Galleries, Observatories, and Centers, but the Institution has only one Institute.