I have an uber awesome amp, why buy a speaker with a built in woofer sub-woofer amp

I have main speakers that are 3 db down at 20 hz.  I added a couple of powered subs that go to 16 hz for movie watching, but listen without subs to  just music.

A lot of so called full range speakers do not deal well with bass below ~35 hz, which is fine as long as you don't know it is missing.

My 3rd system has Vandersteen 4s that have separate bass amplification.  I believe Richead went to on board amps in subsequent models to ensure quality of bass (people over driving unsuitable amps for the bass)

While onboard powered subs in a speaker can cause problems in optimal speaker location, their voicing, being that the manufacturer has control of that aspect, trumps the separate sub or subs in some senses. One such instance is the ability to integrate the sub into the overall frequency response pattern, a benefit which answers to some degree the main drawback of adding subs to a system. Cabinet resonance must be dealt with and getting the best ball response and imaging from the same location requires careful placement. In my experience the placement of this type of speaker is even more critical than normal. This was brought to my attention by Jim Smith and I must say he was on the money here. 
Nodes are another factor where having the subs in the same location as the mids/tweeters can cause issues. But to be sure dialing in the correct placement of separate subs can be frustrating as well, not to mention speaker cable placement and ability to have the subs in a user friendly location. 
All in all, I worked it out to my satisfaction and wouldn't change anything. The balance in the spectrum built in by the manufacturer overcomes the downsides, at least in my experience.

I've yet to find an amplifier that plays bass as well as the amps I have. So I tend to prefer speakers that have deep bass response. The speakers I have at home go to 20 Hz and are easy to drive- 98 db, and my amps have full power to 2 Hz so I can shake my home easily enough!

But my speakers are expensive and I've no worries if my stereo takes up space in my living room- I'm involved in the audio industry after all so it had better! But that is not the case for all people of course so a power sub can be really handy for keeping the sprawl under control and also getting some decent low end extension.

I do find though that if a sub is used, one needs to exercise a lot of care setting it up, otherwise blend issues will plague the system. It is precisely that which is why I prefer full range speakers, so I don't have to deal with that.

A general rule of thumb is all about a dedicated subwoofer amp may perform better on low frequencies than uber-awesome full range amp.