How is the 2016 Zurich Chess Challenge "new classical" format different from rapid chess? For the upcoming (2016) Zurich Chess Challenge, a new time control will be put in place. According to chess24, 


  Six players including former World Champions Vladimir Kramnik and Viswanathan Anand compete in a "new classical" and blitz round-robin event with a novel format and time control.


which is later clarified as


  A five round-robin tournament in which players receive 40 minutes plus 10 seconds per move for the entire game.


How is this different from a standard rapid time control? And will this section of the tournament affect the players' classical ELO rating?
This is not different from rapid chess and will be rated only for the rapid Elo.

The name "new classical" is part of a push to shorten the classical time controls. Originally the idea was to have a time control with around 60 min per game, but in the end they decided to shorten it a little to make the games eligible for the rapid rating at least.