Lots of people around CC with experience of the various private schools in NYC/Westchester/Fairfield, but it will help focus replies if you give a little bit more info on what makes a “great” private school to you/your student! The two you have named both have great elements, but as @417WHB pointed out they are very different in some meaningful ways. A lot of CCers are thinking largely in terms of college placement, for a very particular set of colleges- there is a long thread on that right now:
But sooner or later it comes back to where your student will shine!