On the 737 the headset eliminates noise from the recirc fan, equipment cooling fan, and most of the pack noise.
Longer flights are less fatiguing with the headset because of the reduced low noise.
I do not feel the need to augment the ANR with earplugs. Clamping pressure of the earpieces is low. The large earpads work well to spread the load around the ear. Using soft seals has eliminated the problem of hot spots. The reduction in ambient noise provided by the ANR makes it easier to hear what the other pilot says, so I keep both earpieces in place all the time.