Neanderthal demise remains an enigma. Bigger, stronger, with more cranial capacity? Large displacements can be explained, but to vanish? With so many unpopulated places on the planet, it is unfathomable that they did not survive somewhere.
Neanderthals were very well-muscled but their average height of a male was 5'6". All signs indicate that despite the larger cranial capacity they were not more intelligent: inferior tool-making; inferior 'artistic' expression; inferior adaptability. If they were more intelligent, by whatever measure, then it seems clear enough that their arrangement of cognitive traits was less effective than that of humans. I would guess that they engaged in far less cooperative behavior and were far less inquisitive (based on certain archaeological evidence, such as a tendency not to travel far from home their entire lives).
Sasquatch.
A small population may have survived into the late 19th century, according to Ivan T. Sanderson, writer of a book (published back in the 60's or early 70's, I think) Abominable Snowmen: Legend Come to Life.
It details stories of the capture of a "wild woman" in a frontier area somewhere around the Ural Mountains, that she was kept and "used", giving birth to halfbreed offspring which did not survive when she took them down to the icy creek to wash them after birth.
Sanderson has a ton of stories of human-like creatures; credible or not, the book is an interesting read. I think I'll check it out a my library again, it's been at least ten years since I've seen it.