Hmm, I don't know why they say that. Ice will conduct heat away, reduce inflammation and relieve pain more effectively than cool water. Maybe they are worried about an open wound becoming infected, or damage to the skin from becoming too cold. If you apply say, a can of frozen juice directly to the skin you can get a mini freeze burn. Ask me how I know this? So use a towel to insulate the ice pack from the skin.
What I'm saying here may not be the universally accepted medical opinion, but I'm saying it because I know it works. If you have a bad burn, cool water isn't going to make you feel a whole lot better, because it's just not cold enough.
How well does it work? To me, the results are almost miraculous. Ice kept my would-be 2nd degree burn limited to just a 1st degree burn. Once long ago I was using a propane torch to heat a knife tang and for some reason the nozzle suddenly popped off. Without thinking, I automatically reached down to pick it up. Well, I ran inside to get ice as fast as I could because I could almost feel an instant blister starting on my thumb. I kept ice on there until the pain level went down, which took about 6 hrs. I pretty much used every frozen item in the freezer during that time. But the next day it never developed a blister and the skin didn't peel.
Another time a family came to see me with their teenage daughter who had a really severe sunburn, she had gone on a boat and been on the ocean all day. Her whole torso was badly sunburned. I told them to use ice, the next day they came back and thanked me. If people come back to say thanks, that tells me it worked really well.
Creams, Tylenol, ibuprofen etc are not very effective, but ice is fantastic. Much more effective than any other thing you can do.