Passive is like insulation. Insulating a room (or headphones) does block out noise.

Active refers to a specific technology. Short ELI5 explanation is that sound waves travel into your ear "active noise cancelling" produces the "opposite" invisible waves and "knocks out" the sound.

If you things are not working with your current headphones, I really do recommend a pair of Bluetooth active noise cancelling headphones because they will block out exterior noise, and you can also listen to some lovely sleepy music. This ought to really drown out the unwanted noise, and replace it with something enjoyable and soothing I enjoy this routine quite a lot, even without noisy neighbors. I wear them to fall asleep as well as other noisy environments such as Walmart, or coffeeshops.

Additionally, for those who do use headphones for music or podcasts, active noise cancelling is much healthier for your ears because you can listen as much lower volumes (since it blocks out exterior noise). This protects your ears from hearing damage over time.

However, as many have stated, other methods of dealing with noisy neighbors such as reporting violations is more of a long term solution than headphones. This is a bandaid... But it can be a very enjoyable bandaid!

You can absolutely use construction or gun headphones if you do find some that work. But imo active noise cancelling is going to bring you closer to your coal. Foam insulation can muffle noise and make jack hammers or lawn mowers down to safe ear volumes, but they don't block out all sound. Active noise cancelling headphones aren't magic, but because they can block AND play quite sound on top of it, I think they would do a better job for sleeping if you are not satisfied with the latter.