Based on the comparing I've done on the second clip so far, they seem to be doing absolutely nothing special - at approximately the same bitrate and settings, the videos are practically identical quality-wise (in fact, the most recent x264 seems to fare a small bit better).

I'm going to do two sets of comparisons for each clip: Beamr's video compared to x264 with similar settings and bitrate, and then Beamr-like settings and high quality settings compared at a much more realistic bitrate that you would actually see in use in digital video on the internet.