If you take too many chances in the process of making a living, you’ll get yourself killed before you manage to raise a family. Therefore there is a maximum sustainable risk per calorie acquired from hunting *. If the average member of the species incurs too much risk, more than that sustainable maximum, the species goes extinct. The Neanderthals must have come closer to that red line than anatomically modern humans in Africa, judging from their beat-up skeletons, which resemble those of rodeo riders. They were almost entirely carnivorous, judging from isotopic studies, and that helps us understand all those fractures: they apparently had limited access to edible plants, which entail far lower risks. Tubers and berries seldom break your ribs.
In Africa, most calories probably came from plant foods back in the Middle Stone Age, as they do in African hunter-gatherers today, and that fits too: early African hunters seem to have mainly gone after relatively safe prey like eland, avoiding really dangerous animals like cape buffalo. This is not to say that they did not hunt, or that hunting was unimportant, but they had alternatives.
Risk per calorie was particularly high among the Neanderthals because they seem to have had no way of storing meat – they had no drying racks or storage pits in frozen ground like those used by their successors. Think of it this way: storage allow more complete usage of a large carcass such as a bison, that might weigh over a thousand pounds – it wouldn’t be easy to eat all of that before it went bad. Higher utilization – using all of the buffalo – drops the risk per calorie.
You might think that they could have chased rabbits or whatever, but that is relatively unrewarding. It works a lot better if you can use nets or snares, but no evidence of such devices has been found among the Neanderthals.
It looks as if the Neanderthals had health insurance: surely someone else fed them while they were recovering from being hurt. You see the same pattern, to a degree, in lions, and it probably existed in sabertooths as well, since they often exhibit significant healed injuries.
In the recent extinctions, hunting megafauna must have been relatively safe. The target species has to be attractive (a good food source), safe and easy enough to hunt, and have a low enough reproductive rate, if they’re going to go extinct. Rabbits are not very attractive, since they furnish a small amount of meat. They’re safe to hunt, but not that easy to catch, and they have a high reproductive rate – so they’re not going to go extinct. On the other end of the spectrum, giant ground sloths furnished a lot of meat, and were easy to catch. They were probably easy and safe to hunt using distance weapons like the atlatl, since they were slow. They had a low reproductive rate. Prime candidates for extinction.
So we can often understand the pattern, but why were mammoths rapidly wiped out in the Americas while elephants survived in Africa and south Asia? I offer several possible explanations. First, North American mammoths had no evolved behavioral defenses against man – while Old World elephants had had time to acquire such adaptations. That may have made hunting old world elephants far more dangerous, and therefore less attractive. Second, there are areas in Africa that are almost uninhabitable, due to the tsetse fly. They may have acted as natural game preserves, and there are no equivalents in the Americas. Third, the Babel effect: in the early days, paleoIndians likely had not yet split into different ethnic groups with different languages: with less fighting among the early Indians, animals would not have had relatively border regions acting as refugia. Also, with fewer human-caused casualties, paleoindians could have taken more risks in hunting.
* this is an oversimplification. The sustainable risk varies with age, sex, etc.