If a Speaker or Subwoofer is well designed and capable of producing high quality sound, why does it matter whether that sound is "music" or "HT"?
I cannot speak to those two subwoofers, because I am not familiar with them.
However, in an effort to justify a differentiation between music vs HT usage, I can say for myself, with any sub that I have setup, I prefer to run my sub hotter for HT (action movie, not music movie) than for music.
I'll follow this with some discussion, but whether this is personal idiosyncrasy, reaction to contrasting production values, or any of many other possibilities, it does establish that I do not want my HT and Music sub to be identical in every way!
The hotter HT setting sounds a bit muddied for most of the music I enjoy, but I could also argue that most music written for the movies is not so sensitive to the extra bass! It doesn't bother me to have that extra bass weight for the Theme to StarWars as it just adds some "epic weight" to it - same for soundtrack for Avengers ... but Melody Gardot or Jaco Pastorius? No thank you!
In my lazy one-setting-fits-all scheme, I decrease the bass setting for music inputs by 1-2dB from where Denon/Audyssey setup would have it. Some music is fine if I left it at 0, but other music is bass heavy at 0dB! It depends on the recording! For music, it is
very important to me that the sub never stands out by being over-enthusiastic!
Conversely, I increase the sub level 4-6dB (IIRC) for HT! I don't mind it at all when my subs take the stage and impress me during a movie! It is fun! Of course, the target for a movie cannot be accuracy, because for an explosion that involves concussions and structural failures!
Also, looking for reasons why music and movie objectives might be different, the LFE is an obvious reason. Generally, music is produced under the oversight of the musicians (ideally simply using an engineer they have come to trust to make it good). I don't know, but I might presume that movies have standards and guidelines on appropriate use of the LFE channels and the LFE is more about tactile sensations than (most) bass is for music.
Having said all of this, My Outlaw X13's are the first real* ported subwoofer that I have heard without being a liability for music. I believe that is because the the overall design/quality is not prone to being sloppy and because it has enough tuning options to allow me to adjust to the sound I want.
I can't remember what subs you had previously, but there are many subs at lower prices where you do not have this level of adjustability (unless you add it via tools like XT32 and/or miniDSP). In these lower price ranges, you are often having to decide between compromises that provide better quality and output at frequencies above ~30Hz for music vs tactile feel for HT! If you are not going to use Audyssey or miniDSP, you have to be very careful picking a subwoofer to make sure the compromises are consistent with your objectives.
For me, that means buying a sub for music because I can handle not having the deepest or loudest bottom end in a movie, but I cannot handle having deep bass that is overwhelming while listening to music.
Since you have the X-13 which is nicely capable, I can appreciate that you are familiar with a sub that does both very well... however, I hope you can see how - at least on a budget - the preferred compromises would be different between a music vs HT sub.
* By "real" ported subwoofer, I mean a sub ported to maximize extension/SPL. For example, the Infinity R12 ($200, when on sale)is ported, but it is tuned to play mid-bass efficiently (I would guess the tuning frequency around 38Hz based on FR). I believe Infinity intended this sub for music. This sub would be very much deficient for HT, but makes a wonderful
subwoofer for music to use with mains that have 6" woofers. Looking at the FR, I wonder if Harmon/Infinity may be deliberately attenuating the bass to allow some leeway for Room Gain before it gets overbearing - surely they could have tuned it to be flat at 40Hz if they wanted - with Harman/Infinity, I don't think they "accidently screwed up" the FR.
Of course, compared to TV/soundbar speakers, this is a great subwoofer for HT!