How is LMS/FXLMS noise cancelling different than simple polarity inversion? Consider a noise-cancelling headphone:

If I have a noise signal from the outside world, mic it, flip the polarity by multiplying by -1 (with an op-amp or digitally), delay by the appropriate amount, and playback out of the speaker -- the summation of the noise and anti-noise at the ears should approach zero (with better performance at low frequencies).

How does FXLMS or LMS improve upon this? Why is polarity inversion ('phase cancellation') not as effective at cancelling noise, at least in the literature (e.g. Kuo 1994)?
Your polarity inversion method simply applies the same phase shift (delay) and gain to all frequencies, and is presumably not adaptive (or even closed loop). It may work reasonably well for some limited cases, where the phase and gain have been tailored for that particular environment, but any change to the environment will tend to reduce its effectiveness.

LMS however is an adaptive method which constantly "tunes" a filter to give you optimum cancellation using an appropriate gain and phase response which varies with frequency, and which adapts to a changing environment.