Small differences, it is the same story as before. The theme music has been changed, special effects improved upon, the nudity removed, a few scenes were added and existing ones altered slightly. One example of this last point would be an early scene, the one during which General Hammond is showing Jack O'Neill a Goa'uld staff weapon. The line "we can't figure out how it works" just before O'Neill expertly activates the weapon was removed. Minor changes, in other words. A detailed comparison, split into two parts, with pictures can be found here (Part 1) and here (Part 2).
No, the nude scenes in the original Showtime episodes have been edited. The nudity is now implied by bare shoulders and backs. USA movie theater rating would probably be around PG (or PG-13) and the DVD could be showed "as is" on American television without any changes.