There is no evidence that Neanderthals had fur or looked as the picture you posted looks. In fact they stood upright, made tools and lived hundreds of thousands of years before dying out. They have been classified as human, with Human DNA, slightly different than ours. If they were furred that would be evident in their remains. It isn’t. There are Humans on this planet now that share some Neanderthal DNA. I am sure there would have been no mating between them if they looked like the ape you have posted.
It pays to keep up:
In brain size Neanderthals were comparable to modern humans or even slightly larger--the Neanderthals ranged from 1300 to 1700 milliliters whereas modern humans range from 1200 to 1400.
There is a recent scholarly (but very readable) book on the Neanderthals, originally published in German in 2005, English translation 2009: The Neanderthals, by Friedemann Schrenk and Stephanie Mu+ller, translated by Phyllis G. Jestice, published by Routledge (paperback ISBN 978-0-415-42420-9). No National Geographic-style color photos but lots of interesting black-and-white drawings and maps--a very interesting book with up-to-date information.
The Neander Valley, by the way, is named for Joachim Neumann, a Bremen theologian and hymn-writer of the 17th century (composer of "Praise to the Lord, the Almighty, the King of Creation"). Neumann means "new man" and "Neander" was coined from the Greek roots neo- and andr- meaning "new man."