In August 2014 an international research team reported (in Nature) 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 - 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
No theory as yet has has been universally accepted by experts in the field. The explanation could be a combination (or none) of the above.
Further reading Wikipedia