There is enough Neanderthal DNA in the modern non-sub-Saharan human population (we all have a few percent Neanderthal DNA, but that few percent is different in each of us) that some people probably think like Neanderthals.

I was more talking about the different mindset ... coming from a very different culture.

My point was actually that the DNA difference probably does not lead to a very different thinking. But different culture does.

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.