Using headphones with active noise cancellation has a lot of advantages, making it hard to go back once you get used to them. The most obvious benefit is getting rid of annoying sounds; it's hard to imagine a long plane trip without noise cancellation. Because you don't have to overcome outside noise, you also enjoy better audio quality without distractions, hearing more details in songs or movies. That also makes ANC headphones safer for your ears; you won't need to turn the volume all the way up (and risk damage to your hearing) to drown out the outside world.
It's also great for phone calls, as you'll hear the other party more clearly. In some cases, where you don't even want to listen to music or watch anything, ANC can still be useful for simply providing silence when you need to focus. It's especially helpful for people who react negatively to external stimuli. Some people even wear ANC earbuds to sleep.
Of course, not everything about ANC is perfect. ANC works best with low, steady noise like the hum of a plane or air conditioner. It's not as effective against sudden higher noises, like horns honking or people shouting. Depending on the exact headphones, ANC can sometimes reduce audio quality, and some people say their ears hurt when using the feature.
Additionally, Isolating yourself too much from the world can become a problem: without hearing the world around you, you might become too distracted in situations that require extra attention. For instance, hearing the sound of cars as you cross the street or ride a bike is crucial. Talking to nearby people with noise cancellation turned on is also terrible. Some studies even suggest that noise cancellation may affect how your brain filters out background noise. That's why many headphones have a hybrid mode — like the Transparency Mode on AirPods. These balance ANC with sound from the outside world.
There's also the fact that noise cancellation drains the battery much faster. Turning the feature off can help your earbuds last longer throughout the day, though many people are willing to pay this price for silence.