Difference between these two digital down conversion methods
As you already know, the two methods are theoretically identical if there's no noise. If there is out-of-band noise then the low pass filters in method A will further suppress the out-of-band noise, whereas there's no such noise suppression with method B. This would be one advantage of method A over method B.

Note, however, that the phase splitter in method B (which is the complex-valued filter with a wire in the real part and a Hilbert transformer in the imaginary part) could be replaced by a complex valued band pass filter filtering out the positive in-band frequencies. With such a filter, both methods would be equivalent, even with out-of-band noise.

I do not think that the need to implement a Hilbert transformer is a disadvantage of method B, because the Hilbert transform does not need to be approximated over the whole band, just over the signal bandwidth.