The burn is on the surface of the skin, where there are lots more nerves in the first place. As mentioned, in a light burn the nerves are damaged not destroyed. So you’ve got more pain sensors for hot and cold, more nerves per square inch, and they’re not cut or destroyed = more pain.
A cut doesn’t hurt as much because if it’s a surface cut (paper cut) then you only hit some nerves and sever some as well, but you don’t have the double-whammy of the hot + cold sensors in the nerves shouting as well. If it’s a deeper cut, you only really have the surface nerves, some of which are cut as well, and you have fewer and fewer nerves the deeper you go.
A few years ago when that nightclub in Rhode Island caught fire, I read an account of one of the doctors who treated people who were taken to a nearby hotel (I think). He said that when determining who to treat first, he ignored the people who were screaming in pain, because he knew they were “ok” (in general). He was more worried about and looked for and tried to treat the ones who were awake but acted as if they weren’t in any pain. He said he knew they were the ones in the worst shape and probably close to death. That opened my eyes to the importance of pain (or lack of pain, indicating severe nerve damage) in burns.
Full anesthesia is not perfectly safe (which is why a doctor is needed to calibrate it to each patient). In rare occasions, the patient can slip into a coma or die.
One year my mom made a fire in the fireplace at the house. There was a lot of debris in the fire place and I have no idea how this got left in the fireplace, but there was a spraypaint can in the fireplace with the lid on fire. At the time I was about 9 years old and bending down in front of the fireplace I saw the paint can on fire and said, “Ma, this spraypaint can is on fire!” and she said, “What spraypaint can?” Anyways, long story short, my Dad got a pair of long crab tongs and tried to get it out. He told us to run and as soon as he said that, BOOM. It exploded.
Everything just blew up in the kitchen, including my dad. My mom and I had just gotten out of there before it exploded, but not my dad. He got the brunt of it unfortunately.
Anyways, he had no hair, no eyebrows, no eyelashes, no skin on his face. Luckily there was snow outside, deep snow, in fact we were in the middle of a snowstorm. He ran outside and lay his face in the snow, but that only helped temporarily. So we had to drive him to the hospital in a Jeep. I have to say, I’ve never seen my dad in so much pain, it was really really horrifying to see that as a young boy.
The doctors pretty much told him that a burn is one of the worst injuries you can get. And as far as the answer to the OP, someone pretty much summed it up above when they said that the nerves are injured in a cubed fashion.
It took my Dad a long time to recover, but he eventually did, but not without a lot of morphine and silverdine. Man, I don’t wish a burn on my worst enemy, it’s traumatic. I feel for burn victims.