Would a too powerful sub for quiet music degrade sound quality?

[Pyrrho]

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Also the answer to this particular question may be answered by knowing if the driver used in these two enclosures are the exact same? If so, then they only change the amp and change the enclosure size slightly. This would indicate it wouldn't matter at low listening volumes since the driver would still be moving exactly the same.

This is wrong. The different cabinets likely affect the frequency response of the different subwoofers, so the better one will likely go deeper than the other one when both are at the same volume at the upper bass, even if the actual woofer is the same.

I personally recommend using an unpowered subwoofer with a separate amplifier (see the reasons for this in my post above to someone else).