How is the ratings R and NC-17 different?

Answer 1

The MPAA defines them as:

R - Restricted. Children Under 17 Require Accompanying Parent or Adult Guardian. An R-rated motion picture, in the view of the Rating Board, contains some adult material. An R-rated motion picture may include adult themes, adult activity, hard language, intense or persistent violence, sexually-oriented nudity, drug abuse or other elements, so that parents are counseled to take this rating very seriously. Children under 17 are not allowed to attend R-rated motion pictures unaccompanied by a parent or adult guardian. Parents are strongly urged to find out more about R-rated motion pictures in determining their suitability for their children. Generally, it is not appropriate for parents to bring their young children with them to R-rated motion pictures.

NC-17 - No One 17 and Under Admitted. An NC-17 rated motion picture is one that, in the view of the Rating Board, most parents would consider patently too adult for their children 17 and under. No children will be admitted. NC-17 does not mean "obscene" or "pornographic" in the common or legal meaning of those words, and should not be construed as a negative judgment in any sense. The rating simply signals that the content is appropriate only for an adult audience. An NC-17 rating can be based on violence, sex, aberrational behavior, drug abuse or any other element that most parents would consider too strong and therefore off-limits for viewing by their children.

Answer 2

If you're under 17, you can see an R-rated movie in theatres if an adult buys your ticket and sees the film with you. No one under 17 is admitted to an NC-17 movie, not even with an adult.