According to a recent study

No bows were discovered there, just to be clear. Bows and other wooden objects don't preserve very long. The earliest unbroken bows were discovered in bogs in northern Europe around 11,000 years ago.

Earlier stone and bone point findings imply that bow-and-arrow hunting technology first appeared in Africa between around 80,000 and 60,000 years ago. Moreover, previously discovered fossil teeth show that H. sapiens arrived to Grotte Mandrin as early as 56,800 years ago, significantly earlier than previously assumed and much before the extinction of the Neanderthals around 40,000 years ago.

There is no proof that Neanderthals used this technology in Europe at that time. It's also questionable whether archery gave humans any significant hunting advantages over Neanderthals' spears.

At Grotte Mandrin, 852 stone artifacts dating to around 54,000 years ago, including 196 triangular stone points, showed signs of high-impact damage. 15 additional stone points displayed evidence of both high-impact damage and modifications brought on by butchery tasks like cutting. Damage on stone reproductions that the researchers employed as arrowheads fired from bows and as the tips of spears placed in portable throwing devices was compared to those discoveries. Arrowheads made of stone and bone that were utilized by modern and ancient hunting cultures provided more comparison data. Stone points from the French site exhibited impact damage along their edges, proving that they had been affixed to shafts at the bottom.

According to the researchers, the smallest Grotte Mandrin points, many of which had a maximum width of no more than 10 millimeters, could have only pierced animal hides when fired from bows as the tips of arrows. They discovered through experiments using replicas of the ancient stone points that only arrow shafts shot by a bow can push stone points smaller than 10 millimeters in width to useful hunting speeds.

The researchers suggest that larger stone points, some of which were many times the size of the smaller points, could have been arrowheads or spear tips that were hurled, thrust, or launched from portable spear throwers. The first H. sapiens at the French rock shelter may have used both spears and bows and arrows, depending on the terrain and prey they were after, as earlier research revealed that between 70,000 and 58,000 years ago, sub-Saharan Africans also rotated between these two kinds of hunting tools.

H. sapiens who were recent immigrants to Europe may have learned from Neandertals that spear hunting in big groups is preferable in cold climates where bow strings can easily break and long-distance animal pursuit is not energy-efficient. Yet it's possible that Neandertals were not meant to learn archery from Homo sapiens. Based on past investigations of brain imprints on the inside surfaces of fossil skulls, some suspects that Neanderthals' brains did not enable the heightened visual and spatial abilities that H. sapiens utilized to hunt with bows and arrows. I personally doubt that. I do not think that’s a fair conjecture to make.

See, more and more evidence contends that Neanderthal behavior was identical to that of Stone Age Homo sapiens. Although though Grotte Mandrin Neanderthals may have never used a bow and arrow for hunting but yet coexisted peacefully with H. sapiens archers for around 14,000 years, the causes of Neanderthals' final extinction are still unknown.