Fascinating reading everyone's take on this, particularly N90-EWR's.
There's a lot of talk about Captain's authority etc, and in the clip I see no evidence of this being undermined. I agree that there were comments from both parties that were clumsy and unnecessary.
The point I would make, however, is that a lot of the conversation above revolves around what happened in the air; To use common TEM parlance - how the threat of weather (and mid-air cillision) were mitigated against. But this authority of the commander does not magically appear on rotation - it exists on the ground. So I wonder if maybe the whole thing could have been avoided. NY TRACON can't make the aircraft turn left, just as JFK TWR can't make the captain depart.
Was the left turn after departure what the crew were expecting? If not - was their mental model deficient?
I don't have the pleasure of visiting the NYC area, but my home base is also in a busy, although probably simpler TMA. As far as I have been told, if we feel we can't comply with our departure instructions due weather, the tower would rather know in advance and would be happy to coordinate a safe route with the departure controller. What they really seem to like is if you tell them what you want/need clearly, and concisely.
Just my two penneth.