Contagious diseases did not arise (measles, mumps, chicken pox, tuberculosis, etc.) until after the domestication of animals. This did not occur until well after the disappearance of Neanderthals. Therefore, disease was likely not the reason.
There was some evidence of interbreeding, yes, but not enough to continue the dominant characteristics. Mitachondrial DNA would probably be needed to substantiate this.
There have been a mix of fossils found in caves, but whether this meant co-habitation or successive living arrangements (one group, hss, displacing the other) is being debated.
I think there is much to be said for the reasons for the demise of the Neanderthal being related more to the change in climate making living for the Cro-magnon more possible, for the different abilities that Cro-Magnon had which allowed for finer tuned hunting, more sophisticated diets and clothing, technological changes that could be built one upon the next, and the time for leisure and the development of decorative items and drawings. That much is reasonably certain.
For neanderthal as well as Cro-magnon there is evidence of burial indicative of some spiritual belief system.
As for speech, this is a point that paleolinguists are still debating though more credit is being given now, I think, to the Neanderthals having more capability than was first thought.
Survival of the fittest would be an explanation but with a change in habitat due to climate being a factor in 'who was fittest'.
