Actually my thread had them making it to the New World. I also did one where they survived in Britain.
From what I've pieced together, the Neanderthals were specialized for hunting large game through ambushes in forested areas. Based on the number of broken bones, they got up close and personal with their prey--the only modern human population with comparable number and type of fractures is rodeo riders.
They were far more muscular than modern humans, but probably nowhere near as good at long-distance running. They probably covered smaller territories than most modern human hunter-gatherers. They were probably not well-suited to open plains. Apparently living on the open plains required too much long-distance movement to follow the game, and Neanderthals' bulky bodies wore out too quickly in that environment. Open plains were kind of emergency use only areas, and the Neanderthal skeletons from those areas showed signs of crippling arthritis. Joints just wore out.
Again, piecing things together: I'm guessing that Neanderthals died out as a result of a serious of events: (1) A series of volcanoes (smaller than Toba but regionally devastating, nearly exterminated the populations in the Caucasus and surrounding areas, allowing modern humans to flow into open plains north of the primary Neanderthal area. This wasn't prime Neanderthal territory anyway, but the moderns were better than Neanderthals at exploiting the open areas and that cut the Neanderthals off from using the plains as a buffer when forests shrank in front of the glaciers. (2) around 40,000 years ago a couple of regionally devastating volcanoes knocked Neanderthal populations in Italy and surrounding areas down low enough that moderns just swamped them. (3) Neanderthals held out in southern Spain for up to 10,000 more years, but the last advance of the glaciers essentially eliminated suitable (forested) habitat and they died. (4) Recent genetic studies seem to prove that a percent or two of the DNA for Europeans came from Neanderthals, though the interbreeding seems to have happened exclusively or nearly exclusively in the Middle East before moderns invaded Europe.