Reddit mentions of The Humans Who Went Extinct: Why Neanderthals died out and we survived

Not fiction... but if you find the era interesting:

• Sapiens
  
• The Humans Who Went Extinct: Why Neanderthals died out and we survived
  
• Before the Dawn: Recovering the Lost History of Our Ancestors
  
  Jean M Auel provided a fun introduction to the period but, as readers of the series know, she took creative liberty. More than occasionally the books wandered into fantasy. To enjoy a fascinating view based a bit more on scientific evidence, the three listed are (or were to me) quire enlightening.

Well, that's the million dollar question really. Some like Clive Finlayson argue that it was that humans stumbled upon a cultural package that led them to genetic victory or others say that modern humans could have simply been slightly more fertile than other hominins and replaced them over time without conflict.

However, the fact that modern humans seems to have displaced all other hominin species wherever they have found them remains. It's quite surprising that we only have 2-4% of other hominin species in our DNA; the fact that some ancient humans had like 20%+ Neanderthal DNA indicates that such DNA had been selected against.

And there are genetic differences we posses with Neanderthals,
a interesting preprint (currently undergoing peer review) has found that modern humans have undergone much selection on the face and the vocal tract. The trait that cause our faces to recede unlike Neanderthals seems to have consequences for the evolution for our ability to vocalize; so there also could have been substantial differences that way.