Amazon would not pay less NYC tax than other businesses. The NYC tax breaks that Amazon was planning to take advantage of are open to every company that qualifies:

https://www.nycedc.com/program/relocation-and-employment-ass...

https://www.nycedc.com/program/industrial-commercial-abateme...

These programs are designed to encourage specific behavior, like companies moving to and making capital investments in particular areas of the city. Any company that does so qualified for a formulaic tax break. NYC did not offer any tax breaks to Amazon beyond its existing incentive programs.