Though about 90% of Mafioso was shot in Italy, there was some location filming in NYC toward the end. It was a little strange seeing some parts of New York (not necessarily the "touristy" areas), in gritty black and white, that have long since vanished. I certainly remember those areas, or areas just like them. Brought back some memories, to be sure.
I believe the barbershop scene was filmed in North Bergen NJ. TCM showed "Mafioso" for the first time as soon as the DVD was released, so I assume the version you taped and the DVD are one and the same.