We certainly know that different DNA leads to different thinking or else we would think like a chimpanzee. The question is does Neanderthal thinking live on in our modern culture. I have no idea, but it is a testable hypothesis.
Surely does different DNA leads to different thinking. But when interbreeding is possible, differences might not be great ... much smaller than the cultural differences
That is a hypothesis, not something we know. Having three children of my own all very different to each other in thinking makes me doubt culture has much influence on how people think.
My point was actually that the DNA difference probably does not lead to a very different thinking. But different culture does.