Ear-plugs work by blocking sound pressure; good ones by up to 21dB. Active noise-reduction headsets work on cancelling out sound/noise within a specific frequency range - the sound is picked up thru' the boom-mic, the noise-reduction circuit generates an out-of-phase signal, and you then hear 'nothing' in that frequency range!!
Apart from cancelling out some of the ambient noise, ANR circuits 'clean' out the inherent noise in the radios (HF reception is almost as bearable as VHF). What this means is that you can have the receiver volume turned down lower and you can hear your other crew members speak.
Try it out for yourself, and you'll find that ANR headsets have loads more advantage over ear-plugs (but ear-plugs are cheaper, much, much cheaper...)