First things. Look at those 2 skeletons again. Notice the funnel shaped chest cavity on the Neanderthal compared to the barrel shaped chest cavity on the modern human. On the Neanderthal the hips are wider, to match the bottom of the chest cavity. Look at the thighs. The Neanderthal thighs slope inward to the knee at a much sharper angle than the modern humans. This is required by the broad hips to keep the feet under the center of the body. Look at the forehead. The Neanderthal forehead slopes back sharply from the brow ridge while the modern human forehead is almost vertical. The Neanderthal features are far outside the normal range of modern humans.
The Neanderthals do appear to have been much stronger than the average modern human. This appears (according to people who have studied skeletal development) to be partly the result of the fact that Neanderthals, on average, worked much harder than modern humans. Hard work develops very strong bones. This is probably a cultural trait. The Neanderthals apparently did not develop cultural responses to the environment that enabled the modern humans to live without working as hard.
The Neanderthals had larger brains than modern humans. Unfortunately the relation between brain size and intelligence is still not clear. Intelligence does seem to increase when the brain/body ratio increases. I don't have any good numbers right at hand, but let's just say that we have 2 animals that are the same size. One of them has a brain that is 5% of the body size. The other has a brain that is 2 1/2% of the body size. That's just picking some numbers out of the air. In any case we could expect the 5% animal to be smarter than the 2 1/2% animal. However, this is an average thing. Within our own species some people who are definitely highly intelligent have smaller brains than some people who are definitely not very intelligent. So we can't really say how intelligent the Neanderthals were based on brain size. We can say that they probably weren't particularly dumb. It may just be that their talents went in a different direction.
What does it take to survive an ice age? Well, for one thing it takes enough smarts to come in out of the cold. In order for the Neanderthals to have survived the ice ages they must have had shelter, fire and clothing. Then they needed a way to get food. Those are probably the main things they needed. And of course there is the possibility of going to Florida for the winter. The Neanderthals did apparently move at least to Southern Europe. They didn't actually live with the glaciers. But it did get cold. At the same time this meant that there were large mammals in the colder areas where they lived. So there was plenty of meat to be had, by hunting and scavenging. In a very cold climate scavenging of animals that died a natural death could work quite well, because the cold would preserve the meat for some time. The jury is still out on how much they hunted and how much they scavenged. They probably did both.
As far as building long term shelters, they probably didn't bother. They were hunter/gatherers, they didn't stay in one place very long. You only need long term shelters if you are going to be in one place for a long time. That can only happen if you have a convenient long term food source. If you are living on nomadic animals your food source is going to be coming and going and you will have to follow them. You can use the same camp site every year, but that doesn't need to be permanent. It's easier to use portable structures, or replace them every time you come back to the camp site.
One question about how the Neanderthals lived is whether they did any long term planning. There are some people who think they did a lot of rather aimless wandering looking for needed supplies as they went, without any particular plan. What I am trying to say is that they didn't plan ahead to the extent that they would realize that every year their prey animals would be in location A at time T, so that they could plan to be back at location A when the prey would be there. This is what modern human hunter/gatherers do, but it is still a question as to whether the Neanderthals would do it. Some people say they didn't.