That's what I understood as well. In a way, they live on with us because they are part of the same species. We know that we have bred with Neanderthals. Why aren't they just considered a branch of humanity? It's normal for some designs to die out in favour of others. Slightly stocky humans with bigger brains. If you met one, I wonder what you'd think in comparison with, say, a person from another culture?

this is actually an active argument for the reclassification of neanderthals either directly within homo sapiens, or for a new grouping to created to cover all of these hominids. We have to remember that all taxonomies are flawed, useful simplifications layered onto top of a much more complext biological continuum.

True, but if we go extinct, there is nobody left to preserve our genes.