The advantages of Bandgap B is the possibility to adjust the value of the output voltage when
VREF = R4 * [ VBE1/R3 + deltaVBE/R1 ]. You can configure the ratio R4 in order to achieve the desired value of output reference. As a consequence, BGR B can operate with VDD less than 1.0 volts. It is the classical Banba architecture for bandgap with low-voltage operation.
BGR B requires VDD higher than 1.2, and does not permit adjustment of the output voltage. This is the major difference between these two approaches.
In my option, both circuits are able to be trimmed.
There is a difference regarding the mismatch current. In BGR A, it depends on the matching of two resistor (and offset, BJT mismatch, etc) , while BGR B depends on the matching of the current source (MOSFET).
Regards,