These findings always seem to be suggesting that we more or less warred (fought, disputed territory ranges - pick your euphemism) upon them opportunistically over centuries and didn’t lose.
There is a good case to be made that Homo Sapiens were just lucky to be based in the milder climate of Africa, and by chance being more adapted to hunting megafauna due to our body structure. Homo Sapiens are also very inbred. IIRC there are two main genetic bottlenecks that have been detected in our DNA. We almost didn't make it.
FWIW:
It looks like the Neanderthals had javelins as ranged weapons: https://www.mdpi.com/2571-550x/6/1/17
The paper I linked shows that we were in and out of Neanderthal territories (or vice versa) a bunch.
Punctuated equilibrium (is that still in fashion?) suggests that the inbreeding / limited population eras of humanity are when we got a lot of our special capabilities!