Functionally, they're pretty much the same: in almost all municipalities in the U.S., plastic (PEX for water supply, ABS for waste) is allowed for supply and waste lines. Plastic has been allowed in Europe for a long time. Some early plastic piping had problems, but it seems as if the problems have been worked through, and plastic is now allowed nearly everywhere. Where it's allowed, it's used nearly universally, as the materials and labor are much less expensive than for copper.
For a variety of reasons, San Francisco code doesn't allow plastic: for supply, copper is required. I've seen two houses in San Francisco where someone saved a buck by hiring someone who didn't know the local codes, and all the plumbing had to be ripped out and done again. The domino effect of the fallout was severe, for the plumbers, generals, and homeowners. Just make sure your contractor knows the SF codes and has the plumbers who have real experience with copper.