I saw it at the Public, and have seen photos (but only photos) of the tour.
In terms of the set, the Public used a revolve and everything happened on one plane, rather than the platform that we see in the tour set.
The musicians were in the pit at the Public.
The Public's back wall was the standard brick back wall, which I do not believe features those pipes, and definitely didn't have a back door or those light bulbs.
The New York scenes took place in front of a black backdrop, and it parted to reveal the fully furnished home, disappearing into the wings.