Neanderthal man was a separate species from modern day humans that existed in Europe and Asia before modern humans inhabited these areas. However around the time that modern man colonized these same areas as Neanderthal man, they began to become extinct, and disappeared completely from the fossil record around 30,000 years ago. There are many different theories as to their eventual fate, although many of them have yet to be accepted completely by scientists and experts in this field.
Many experts think that Neanderthals might have interbred with the modern humans who began to inhabit the same territory as them between 50,000 and 30,000 years ago. The main evidence of this theory is largely based upon several specimens which have the features of both modern humans and Neanderthals. Similarly it has also been noted that Neanderthal features still appear in some areas, particularly where Neanderthals once existed. However despite this there are also many experts who do not believe in this idea, and the topic remains fiercely debated.
The Neanderthals were also possibly simply out-hunted by the modern humans spreading into their previously untouched territories. The Neanderthals for example are known to have lacked ay projectile weapons, which of course can make hunting a lot easier. If the modern humans hunted more successfully, and at a faster rate also, then the Neanderthals would likely have struggled to find enough food all of the time. Over thousands of years this could well have lead to their population falling and eventually ceasing to exist at all.
Similarly another weakness of Neanderthals was that they had an almost exclusively meat diet, and rarely would have eaten any other kinds of food. Modern man by contrast was already eating fruits, nuts and berries by this point, and would have found it easier to survive in hard times as a result. If for example in a particular area, the large prey species that both Neanderthals and modern man hunted disappeared, then the Neanderthals would have had to either move or starve. Modern man by contrast would have been able to last a lot longer with the absence of large prey species because of their omnivorous diet.
Unlike modern humans, Neanderthals of both sexes would have been involved in hunting the very large prey species that they fed on. Modern humans instead tended to divide the responsibilities in this respect, with women tending to hunt smaller prey and gather berries and plant foods. This small difference would have meant however that modern humans were slightly more successful at breeding than Neanderthals, due to the fact that hunting large prey was very dangerous.
Although Neanderthals were more resilient to damage due to their builds, less females being able to have children meant that population growth would have been slow. Because men can sire multiple offspring at once, modern humans having a lot less of their women killed during hunting would mean that population growth would have been a lot easier.
Around the time of the Neanderthals extinction, Europe was going through vast temperature changes, and was a lot different that it would have been a few thousand years before this. Where before Europe would have been largely forested, it then became desert scrub, and sparsely vegetated. Neanderthals were used to hunting in colder climates, as well as which they were much better at hunting in heavily forested areas than out in the open. Unlike modern man, which would have chased down their prey, Neanderthals tended to rely more on their ability to sneak close to their prey and stab it with a spear.
This difference in hunting styles would have meant that when Europe started to lose its forests, it would have become a lot harder for Neanderthals to hunt as effectively as modern humans. Similarly Neanderthals were built for strength not speed, and probably weren't much good at running after prey, particularly as they also lacked projectile hunting tools. This meant that they were likely ill equipped for hunting in the plains, which is something that modern humans would have been a lot better at.