rabair, if you read your last post, you will see that you have actually repeated what crash has been attempting to explain to you.
If you weren't there to witness the teapot getting broken, you're left with guesses. Some guesses are better than others, mainly due to evidence, but all in all, they're just *guesses.*
The problem with giving you evidence for the extinction of Neanderthals is that we don't have evidence, per se, of the extinction event. There is no smoking gun at this point. Now, we were to find a thick ash layer somewhere on the planet with a date similar to when the Neanderthals died out, that might be a possibility. If we find higher iridium levels instead of ash at that time period, it might have been an asteroid/meteor impact. If we find evidence of severe climate change, that's another possibility.
At this point, not enough is known about Neanderthals to give you specific evidence as to why they died out. And even if a *smoking gun* was found, scientists would still argue about the conclusions. Unless someone builds a time machine, we will never know
for sure, but I think we have some good ideas.
If that's not a good enough answer for you, then either you need to study Neanderthals (in depth) or give up on science altogether.