I don't know, and I don't want to spend a ton of time to understand the theory and implementation schemes. Even if I try, I probably don't have enough knowledge in this field, to really understand the details. So, the best I can understand at the moment is based on what Dirac Live provided on their website, such as the following description, that is very brief and sketchy.
Bass Control - Dirac
It would seem that the short answer is then, Audyssey SubEQ HT use the time alignment method, based on the measured impulse response, whereas Dirac Live Bass Control (DLBC version, that is a paid upgrade) adds phase control. Since there is no way to measure phase directly using a mic and the software, I am pretty sure DLBC's phase correction is also based on impulse response and the measured time delays (just like Audyssey..), but also use the "all-pass filter" technique in their software, to analyze the phase information and hence correction.
There are obviously articles on the internet they go much further into the details, but I haven't even tried to understand any more than I have to. The fact (to me anyway) is that, Audyssey's time alignment method works well enough but DLBC does seem to work better based on out of the box, i.e. without tweaks, than Audyssey's based on my REW/Umik-1 mic measurements, and it sounds"better" to me, though that is highly subjective so no doubt some people might/would prefer Audyssey.
On that note, I can't help but am still amazed by how so many people, seemingly including even Gene and Matthew feel Anthem ARC Genesis (that also now does phase correction) so good. I am amazed because on the objective side, based on measurements, I have not seen evidence that it is better. To me, they performed poorly relative to DL and Audyssey, subjectively, it definitely can sound very good, but not without manual tweaks.
