Science now classifies them as a subspecies rather than a seperate species. I think the truth is somewhere in between and I find the whole idea fascinating as it's so recent.
I read a book in college for a course on human evolution called
Dance of the Tiger. I enjoyed it and it was really easy to read.
It is a fictional story about some interactions between Neandertals and Cro-Magnons during a thaw in the Ice Age.
The author took some artistic liberties to aid in the telling of the story:
The Neandertals were depicted as white-skinned while the Cro-Magnons were dark-skinned.
The Neandertals had limited diction while the Cro-Magnons spoke like angels singing (too the Neandertals).
The Cro-Magnons were very beautiful to the Neandertals while the Neandertals were ugly to the Cro-Magnons (but not too ugly to mate).
The offspring between them were sterile.
What the author set up was a situation where, over time, the Cro-Magnon out bred the Neandertals until there were no Neandertals left.
Since the offspring were sterile, and more Neandertals would want to mate with Cro-Magnon than Cro-Magnon would want to mate with Neandertals, you can see how eventually you'd just run out of Neandertals.
There was a whole story to the book, a 'you-killed-my-father' type thing going on, but around that was the exploration of a potential scenario where competition between Neandertals and Cro-Magnon ended up with the modern humans winning in a way that wasn't through violence and on purpose.
It was all very interesting and a good yet easy read and I'd recommend it if you're interested in the subject. And this is coming from a guy who doesn't read fiction.
Edited by Cat Sci, : changed sub-title