How is LMS/FXLMS noise cancelling different than simple polarity inversion?
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.