Hiya.
Your model has some flipped normals and/or some 2-sided poly's. I don't use sketchup, but I'm guessing it will happily render both sides of a polygon unless you tell it not to (Hexagon does this as well, but I can turn off "render 2-sided" so I can see the problems). That's why it looks fine in Sketchup.
Maybe there is something in Sketchup that can "unify" all normals on an object to be facing one way?