The MTA (Metropolitan Transportation Authority) runs the subways and the commuter lines to LI and Conn. and a few other lines in NYS. The PA runs the Path line, linking Manhattan and a small area of NJ. NJ Transit has a more extensive network linking Manhattan and NJ.
The MTA is years behind London and Paris with subway "nicities". Many cities have waiting times for buses (also MTA in NYC), right now the MTA is experimenting with a few routes--the whole system is projected (when?)
Meaning that the PA (and the MTA also for that matter) have not kept up with times. Studies have shown that far fewer people from Manhattan use the Airtrain that originally planned. One of the original ideas was to get people out of cars and into mass transit to JFK. But time and time again I heard frustrations about the wait times for the Airtrain (in the middle of the day usually) and when the inter airline terminal is taken out of service at busy times (why not from midnight to 4AM).