Difference between Open Loop and Closed Loop Hall Effect Current Sensor
Generally speaking, an open-loop Hall-effect sensor uses the Hall voltage directly to produce its output signal. This has the advantage of being simpler to implement, but can suffer from a certain amount of nonlinearity.

A closed-loop sensor has a coil that is actively driven to produce a magnetic field that opposes the field produced by the current being sensed. The Hall sensor is used as a null-detecting device, and the output signal is proportional to the current being driven into the coil, which is proportional to the current being measured. This method is more complex than the open-loop method, but it eliminates any nonlinearities associated with the Hall sensor itself, since it is being operated at just a single point in its range.