In analyzing the mitochondrial DNA we can obtain from Neanderthals, it looks like many of them had a common, Cro-Magnon female ancestor. Around 220,000 years ago. I found that very interesting.
Disclaimer: Not a professional paleoanthropologist and it should probably not be a topic here,
but to clarify things a bit
Neanderthals: Homo neanderthalensis; appeared around 430 kya and disappeared around
40 kya
Cro-Magnon: AMH (anatomically modern human); Homo sapiens; old designation due to earliest finds in a shelter site in today's France called Cro-Magnon 1; usually not used to refer to homo sapiens anymore
AMH: appeared around
50-55 kya; we have to wait for a bit more to get an expiry date
Many people tend to focus on European human evolution, even though human evolution is a global phenomenon that did start and mostly happened elsewhere. For a further fascinating read about archaic humans like the Neanderthals, I suggest having a look at Denisovans, who seem to be close relatives to Neanderthals and AMH. Probably going extinct 20-25 kya.
But it's true, archaic DNA of both, Neanderthals and Denovians, can be found to varying degrees in today's population.
Edit: On the quoted message replied to: "We" don't know if Neanderthals went for softer facial features more often or not. To my knowledge, there is no evidence for such a thesis to work on or any falsifiable observation for such a hypothesis. Interbreeding was probably the exception, not the norm.