In August 2014 an international research team reported (in Nature 512, pages 306–309 ^{[ paywalled ]} ) on a new analysis of 40 sites in Western Europe - concluding that Neanderthals died out about 40,000 years ago.

There are numerous theories which attempt an explanation of their extinction - including :

• Failure or inability to adapt to climate change
• Competitive exclusion by modern humans
• Deliberate violent extinction by modern humans
• Extinction via interbreeding (hybridisation)
• Natural catastrophe
• Species-specific parasites and/or pathogens.

As yet, no theory has been universally accepted by experts in the field. The explanation could be a combination (or none) of the above.

Further reading Wikipedia

Also see Neanderthal behaviourplugin-autotooltip__plain plugin-autotooltip_bigNeanderthal behaviour

"Almost everything about Neanderthal behaviour remains controversial.

Source : Wikipedia"

Despite very extensive research using the fossil record and remains of artefacts, it's still unknown whether the Neanderthal people :

* Used language