not some, all of us (unless literally an aborigine) of us have neanderthal dna.
they weren't out-competed, we interbred (a lot).
as far as intelligence, they make much better jewelry and tools and so must be smarter (most textbooks would say), but I'd argue they were equally as smart as humans.
new research is coming out showing homo sapiens using the same advanced "leaf-shaped" stone tools that supposedly only neanderthal could make.
their culture might have been more advanced, but a human and a neanderthal would learn new skills and intellectual things at an equal rate.
neanderthals required about 4000-5000 calories compared to a human's 1500-2000.