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.