ANR stands for Active Noise Reduction, where a microphone picks up the external noise and a mirror of the sound is created electronically and played back out to the ear cups. This effectively cancels out the external noise. It can be extremely effective at removing ambient sounds creating a very clear headset for pilots, protecting your hearing from engine noise. They are design to cancel low frequency sound, so speech, engine noise changes and airflow are easily picked up. They are generally smaller and lighter than pnr headsets therefore can be more comfortable.
PNR or Passive Noise Reduction headsets use noise absorbent foam to absorb sound and gel seals along with a more clamped fit to create a seal around the ears to protect from damaging noise. So these headsets are more about sound proofing.
Choice usually come down to personal preference. PNR will give a good level of protection while still keeping the pilot connected to the aircraft. ANR tends to be more expensive because of the technology that goes into them. It may also help to consider the environment that you will be using them in and the level of noise that you are exposed to.