I am skeptical of the claim that we interbred with Neanderthals. There is no irrefutable evidence to support that claim. Not a single modern human has ever been found with a Y chromosome or mitochondrial DNA that belongs to Neanderthals. Neanderthals last shared a common ancestor with modern humans 500,000 years ago. If we interbred, then at least some humans should have mtDNA and Y chromosomes that date back to that age. The most ancient mtDNA found in modern humans date to about 250,000 years ago, and the most ancient Y chromosomes are from West Africans and they date to about 300,000 years ago.
Without irrefutable evidence of interbreeding, the similarities in DNA between Neanderthals can be due to contamination (the DNA of the European researchers contaminating the Neanderthal specimens). There is also evidence that a Neanderthal was killed by a thrown spear. Because Neanderthals evolved in woody habitats, they do not throw their spears, but they stab their prey at close range. The spear that killed the Neanderthal entered the body at an angle of 45 degrees, which is the ideal trajectory angle for throwing spears. That angle allows the spear to travel the longest distance possible. It is perhaps the ability to kill Neanderthals from a long distance that allowed modern humans to kill off Neanderthals. Modern humans did not have to engage in close range combat with the physically stronger Neanderthals as a result.
Modern Olympic athletes can still throw the javelin long distances and they throw it at a 45 degree angle. They inherited that ability from the ancestors of modern humans. It is the secret of our success against Neanderthals in Europe and western Asia, when modern humans migrated out of Africa 60,000 years ago. Modern humans settled in Central Asia for about 20,000 years and then some of them migrated to Europe while others migrated east to northern Asia 40,000 years ago.