There's a few factors at play which dictate this sort of thing
Surface Area
Motor
Internal Box area
A small driver in a relatively small box will have minimal extension (efficiency drops based on the driver's electromechanical nature inside a particular box). To get extension you need to add more power. But if surface area is too small, then you're asking the driver to work harder (excursion) to deliver the same output. The result tends to be higher distortion levels, although there's countless theories on how much distortion is audible among other factors.
Thus, manufacturers have to make different tradeoffs...
Driver displacement / efficiency
Box efficiency
Amplifier power level
So sometimes you need to brute force it with raw electrical power but that isn't a foolproof method. Heat and general mechanical limitations will always come into play and the result tends to be a less clean sound.
There's other factors, but the above outlines it. A good balanced subwoofer has a driver with plenty of surface area and excursion, placed in an appropriately large box, and thus is still sensitive enough that you don't need to brute force it to get output or decent extension. In a sealed alignment you still ultimately need some level of boost though. You'll find that sealed subwoofers generally have more powerful amplifiers than vented subwoofers, though this does not garuntee superior extension or output by any means.
Here's a link that you might want to read:
http://www.salksound.com/wp/?p=56
One way to get high sensitivity and deep extension is to use a horn. Horns... are HUGE.
Another way to get high sensitivity and deep extension is to use an infinite baffle. You need at least four 12" drivers designed for free air loading.
The box is a very relevant part of a subwoofer's design. So is the driver, and the amp is dictated by these things.
In my sub (soon to be subs) I am using a driver with 33mm of fairly linear xmax. Normally a driver with such high excursion capability is going to have very low sensitivity, but my driver has 89db of sensitivity. Why? Because it's an 18" driver!
Take a look at
a)my current sealed box (teal)
b)a larger and vented box (yellow)
c)a smaller sealed box (red)
d)its 12" sibling in the equivalent of my box. (light green)
Each is being fed ONE WATT:
Just imagine how much added power the less sensitive versions (red and light green) need to just be equivalent to what I'm running. Also notice "the hump" in the red box... this means the small box is very likely compromising sound quality with too much internal box pressure.
