Private-property tow: triggered by a private lot operator or property manager when you parked without authorization. Goes to a private yard. NYC caps redemption at $125 for the hookup + $25/day storage after day one.
NYPD tow: triggered by NYPD for violations (no parking, scofflaw, fire lane, etc.). Goes to a borough pound. Standard fees plus any outstanding violation balances.
If you were towed and don't know which, check the signage at the lot you parked in. If it lists a private operator, that's where your car is. If no signage, NYPD.