In all likelihood, Neandertals were a side-branch that were not our ancestors. If you look at Homo heidelbergensis (AKA archaic Homo sapiens) in Africa, they make a much better ancestor to modern humans. They are essentially an advanced form of Homo erectus, with larger brains. The first modern humans appeared in Africa at about 200,000 years ago and descended from H. heidelbergensis. There are a few scientists who hold out hope that Neandertals somehow intermixed with the modern human line, but the anatomical evidence is very weak, and the analysis of Neandertal DNA is starting to foreclose that theory altogether.
The height/stockiness thing may be a reflection of adaptation to climate. African Homo heidelbergensis and H. erectus was tall and slender; probably around our modern height, while Neandertals, in the glaciated European climate, were short and muscular, like today's Inuits and Eskimos.
Basically, Neandertals were replaced by modern humans from Africa completely by about 25,000 years ago. Modern humans had been migrating into Europe from Africa for tens of thousands of years and the Neandertals were ultimately driven to extinction, probably just outcompeted, as the modern humans were quite a bit smarter.
There is quite a bit of literature on the debate - many careers have been spent grinding through the Neandertal issues. PM me and tell me if you want scholarly articles, or broader-interest writings.
