OP (if you are still there...) The atmosphere at big 3 is very different from the atmosphere at TJ. TJ is an excellent public magnet school. But it is still a public school. The physical plant is outdated (except for the science equipment), the teachers are not all the best, and it is a very mixed population -- although everyone there is very smart and motivated.
A Big 3 private school is a private on steroids. Lots of money, lots of brains, not a very mixed group. Their idea of diversity is: the Senators DD goes there and she is Haitian. It is just a different atmosphere. You really should look at both and then make a decision.
But you still need to get accepted and both are difficult, accepting far less than apply. GL.