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
• Used projectile weapons
• Knew how to produce fire
• Conducted burials
• Practiced cannibalism
• Created symbolic artworks

Many other aspects of their culture are completely unknown - see Wikipedia link above.

Also see : Neanderthal extinctionplugin-autotooltip__plain plugin-autotooltip_bigNeanderthal extinction

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 :