It can be amazing the sounds that you hear with a good set of ANR headsets that you otherwise wouldn't hear. One of the most consistent examples is in a caravan (single turboprop) where you can hear what the punters are saying a couple of rows back that you wouldn't hear when you are wearing a pair of passive DCs. In a DeHavilland Beaver (single, piston radial) I end up closing the pilots window completely and pointing the vent away from me because the wind noise becomes far more significant than the engine noise.
I also find that you can turn the radio volume control down and understand radio communications much more clearly.
ANR headsets are most effective at low frequencies. This means they are great for piston aircraft (including radials). They are rather less effective for high frequencies such as wind noise.
The biggest caution with ANR headsets is that if you are using them in a nosier environment than they can cope with they are crap. I regularly fly Beavers which are extremely noisy in the cockpit. Telex Stratus D50's are great providing the ear seals are in good condition (I'm told the D30's are great too). I couldn't tell the difference between a set of DC X-11's and a set of passive DC's in a Beaver. The Bose I tried I might as well have not had on for the good they did in the Beaver. On the other hand in the Caravan the DCs work very well and I'm told the Bose do too.
Personally, I find it easier to hear warning sounds and unusual sounds with the ANR on. ANR doesn't stop the noise - it just reduces it to a comfortable level.