Well, the story's no longer accessible on-line (at least, I can't find it) so I'm going from memory, but:

They were wearing T-shirts, which I bet were 100% cotton, and I'll bet they were wearing blue jeans as well. They'd have been better off stripping naked.

The older brother was active (at first), gathering materials for and building an improvised shelter; the younger brother seems to have sat down and waited. So he had a head start on the hypothermia (or rather, the hypothermia had a head start on him). This activity vs. inactivity may have been the difference between life and death for them. (Another possible factor was that he was lying on the ground, and his brother was lying on top of him. Intentionally or not, it may have been that the younger brother was providing insulation for the elder.)

After building the shelter, it appears the older brother destroyed its value by "pulling the walls down on top of them". I suspect this would have trapped the water and made their situation worse, not better.

The temperature was only 13 F. This equates to -10 C. It's cold, but not that cold; even with the snow, I believe it would be survivable for someone who was physically active.

At 13 F, they would probably have been encountering snow, not wet snow or rain. I suspect the reason they got wet was because they sat still and let the snow melt on their clothing.

Tom Brown describes in one of his books how his "grandfather" once made him and a companion strip down to their underwear and then walk home through the falling snow. Terrified of being frozen to death, they ran all the way. when they got home, they suddenly realised that they were actually very warm. And Larry Dean Olsen once survived a cold night without a fire by running back and forth in the snow until morning.

Finally, when it did become apparent that the younger one was suffering from moderate hypothermia, the only things that could have been done would have been to build a fire, build a better shelter, or get him up and moving around to generate internal body heat.

But of course, if they had known better, they probably would not have gotten into that situation in the first place.