If there is a major world disaster will Mankind survive? In the past our forebears got through environmental difficulties, so there is no reason to believe that this will not continue to happen. As technology becomes increasingly specialized, things we take for granted will be lost. Humanity will take a few steps back or fall right back into a survival-type culture.
The skill to print could end. Story telling could return as a way to pass on culture. Man has always been inquisitive about what lies over the horizon. Experimentation will still be the key to survival. In a major disaster only those species that can adapt will live through it.
Neanderthals were not smart enough to survive. This is despite their brains being larger than humans. The brains of Man obviously operated differently. This could have been the power of imagination. Neanderthals learned more by direct copying of behavior. Humans can deduce answers from information.
Another important factor is present in humans. We have the ability to live in close proximity to each other. Living in cities is a feature of Man. Neanderthals only existed in small groups. They could not "pool" knowledge and make it available to everyone. We develop and control our environment. Neanderthals did not.