It's complicated.
There is a semi recent theory about mass infertility, a "time bomb" thing in our DNA.
Truth is, no other human species survived but us. All the other went extinct. Climate or competition can work for a few of them but not all.
Also, for Neanderthal, a better climate wouldn't have been a bad thing. They would have thrived just like sapiens. Something else happened. For a time we thought sapiens essentially killed them off. Out competition and war. But recent studies have proven close link between the 2, as the flower on grave mentioned. But not only that, there were a lot of exchange between the 2. Sure, probably war to, but a lot of it was peaceful.
Oh and yeah, I'm not fun at parties.