Does sensitivity matter in powered subs?

In a low watt, high sensitivity rig, it seems wrong to hook up a 500+ watt powered sub woofer. I see some subs include specs on sensitivity, but many don't mention it, and there has been little discussion on it that I can find. Thoughts? I am asking both for my own system but also as a general inquiry. Thanks.
Not for a powered sub, whose levels can be matched to the rest as needed.

Higher sensitivity mainly means less power needed to achieve a certain level, but power needed to drive a subwoofer these days is relatively inexpensive and efficient already. Class D amps clearly rule these days I believe for use in subwoofers.

Flat frequency response at actual volume levels across the potential range of operation is important. The flatter down to 20 hz or so, the better in most cases. Many smaller subs low end frequency response will roll off well above that, making them useful for extending smaller monitors or speakers with limited low end extension still, but not so much when used with more full range speakers already.

Are damping specs given for most subs these days? That's another spec that I think would be of interest to help determine the nature of the sound to expect. High damping in general would be a good thing I think?
In addition to what Mapman states, the power requirements for high sensitivity main speakers and a sealed subwoofer are exponentially different. It takes an enormous amount of energy to move a driver reproducing low bass in a small sealed enclosure, no escaping the laws of physics. With the necessity to double power for every 3db increase in volume, you can quickly run up a big power requirement deficit with the particular engineering requirements for a sealed sub. I've seen a number of ported subwoofers with plate amps that have 1/10 the power of sealed subs.
YEp, power=bass pretty much. Still different flavors, but that's the main ingredient. No getting around that. In addition to the demand that higher volumes put on a speaker for bass, power requirements increase EXPONENTIALLY I believe as the target frequency decreases as well....the double whammy!

Basically, little speakers in small portable devices with very modest power amps have greatly limited bass compared to the rest. The biggest, baddest speakers that can go loud and are FLAT as well down to 20hz (powered subs) have the biggest baddest dedicated power amps running them as well.

Yeah, its an old saying, but size (of speaker and amp both) do matter especially so in the case of powered subs.

The more interesting question has always been how much bass do you need and how much are you willing to pay for it? The sky is the limit there these days!