The main difference in philosophy between the two sites seems to be: a) AH typically would present a balanced view, between measured results vs the threshold of audibility and b) feature set that are useful for real world use, c) would include listening test, that obviously is subjective, whereas ASR typically focused only on the measured results.
Besides that, ASR measured more things, such as 32 tones, linearity, filter response and a different presentation of imd etc., whether such measurements are useful or not, such test may reveal more on the overall results of better or worse engineering design and implementation of the dut. AH would measure more than two channel driven output on multichannel devices, whereas ASR typically don't, because their AP apparently could not do more than two channel driven easily, though occasionally he would/could do up to 4 channels by some fiddling (don't know the details of that).
A couple things I believe (just my opoinion) ASR should improve on would include, 1) while the reviewer almost always qualify his harsh comments on anything worse than -90 dB (0.003%) THD+N as "fortunately not that audible", or "not that audible" etc., he should probably tone down some of his comments on the manufacturer's approach, and take on more constructive style so that he could convey the same message that the consumers deserve better engineering and/or quality control via internal testing etc etc.., yet minimize chance of being taken as an just accusation, unfair criticism, without offering constructive criticism that were often buried in the blurbs due to his overall "ton".
I value both sites for their reviews and/or measurements, and would really wish they don't get pit one against the others inadvertently, ymmv..