It might be worth thinking about later encounters – when the Europeans discovered America and Australia. The Europeans came off best because they had the stronger cultural communal knowledge base, which enabled them to build more powerful weapons. It might be a complete accident of history that the discovery of how to make iron happened on one side of the Atlantic rather than the other.
Could we have done better than the Neanderthals because we had a better cultural knowledge base which gave us more advanced technology and allowed us to work together in larger groups, perhaps with some people beginning to take on specialist roles. At the time that language was first appearing the key to having a better cultural knowledge base would be having a more powerful language. Thus it may be that when modern humans first met with Neanderthals we collectively “knew more” – so we came off best – just as Europeans came off better in America and Australia – because they had better technology.
If we look at language as a self-modifying tool there were almost certainly some key “inventions” – such as being able to differentiate between the past, present and future, counting, etc. Perhaps it was just an accident of history that one of our species, rather than a Neanderthal, made the first key technical advances which allowed language to develop and that Neanderthal brains were just as capable in that respect as our own.