According to the NY State Education Department website:
State-funded programs and services include: Health Services, Homebound Instruction, Textbook Loan Program, Computer Software Loan Program, School Library Materials Loan Program, Transportation (students must reside within 15 miles of the nonpublic school), Special Education Services, Dual Enrollment Programs, and Mandated Services Reimbursement.
So these are the things that are paid for. However, not everything is given to the school. For example, Special Education may be offered in the Public school system. The reasoning is that in the public school, they already have the necessary people on staff, and it won’t cost them much if another kid comes in. Private schools tend to have less need, and it would cost far more than it does in public school. Busing is another good example. They provide transportation, but it doesn’t have to be door-to-door service. They can require centralized bus-stops to cut down on the costs. This may mean some kids have to walk a bit to a bus-stop, and a parent may need to walk with them and supervise the bus-stop until the kids are picked up.