One Biblical passage which is used to support the analogy of The Nephilim to Giants is  Numbers 13:33 ---  "We saw the Nephilim there (the descendants of Anak come from the Nephilim). We seemed like grasshoppers in our own eyes, and we looked the same to them."    However when taken in its proper context, the passage is actually a fabrication / a bad report spread among the Israelites .

 Numbers 13:30 to 33    30Then Caleb silenced the people before Moses and said, "We should go up and take possession of the land, for we can certainly do it." 31 But the men who had gone up with him said, "We can't attack those people; they are stronger than we are." 32 And they spread among the Israelites a bad report about the land they had explored. They said, "The land we explored devours those living in it. All the people we saw there are of great size.

 

Although, I do believe that on the Earth there once were Giants , the Biblical writings do not support this theory to the degree that some would writers would have you believe.  Another passage used to support the Biblical Giant synopsis is Genesis  6:4 which in the King James version of the Bible states " There were giants in the earth in those days " . On a previous page I have demonstrated that this is a mistranslation , and I do believe the consensus among most scholars of both Christian and non-Christian persuasions supports my argument. [See Biblical Giants]

Although the Nephilim were not the ridiculously tall Giants one at first envisions, they were huge in the sense of their body mass and capabilities. The Legends of them are extensive given the distance in time we are attempting to draw upon.

Legendary accounts, are, without exception exaggerations, and misinterpretations. The fact remains however that many ancient trans-cultural middle Eastern texts , both Biblical and otherwise speak of the Nephilim / Sons of Anak who were men, but not ordinary men. They were "the heroes of old, men of renown" as per Genesis 6:4. 

Every time a tale is retold it changes just a bit. Legends have a strong tendency to distort relevant facts, but in most cases the underlying foundation of the story has some basis in reality. The human tendency is to remember and retell things that make a strong impression and to forget things that don't impress us much at all. There is also a tendency to exaggerate , glorify , idealize or vilify. As well as a natural tendency to simplify or edit. After generations of stories being retold of these huge men with exceptional strength, these extraordinary men eventually become giants in height as well as mass and intelligence. They become the Neanderthal of "Collective memory" which the Legends have labeled Nephilim, anakite, and Rephaim giants - and that which science has labeled Neanderthal and possibly Cro-Magnon.

Neanderthal Intelligence

One great enigma of the Neanderthal is their brain size, they had a greater Brain to body weight capacity than modern Homo-Sapiens. Neanderthal brains were 13% larger than our own brain size, The average brain size is larger than the average modern human by almost 200 cubic centimeters. . This cranial capacity allowed for large cerebral hemispheres. Neanderthals were intelligent thinkers. However, despite their probable higher mental capacity archeologists claim that they left no discernable permanent record of their presumed intelligence or culture.

Assuming the Neanderthal and the Nephilim are one and the same, they left an indelible impression upon Human History. They were kings and rulers of many lands in antiquity.  Large segments of the The Old Testament recount a vicious bloody murderous genocidal conquest of the "promised Land" by the Hebrews. Og the king of Bashan was one of the Nephilim. Deuteronomy 3:4 At that time we took all his cities. There was not one of the sixty cities that we did not take from them—the whole region of Argob, Og's kingdom in Bashan. ... 6 We completely destroyed them, as we had done with Sihon king of Heshbon, destroying every city—men, women and children. ... 3:11 (Only Og king of Bashan was left of the remnant of the Rephaites. His bed was made of iron and was more than thirteen feet long and six feet wide. .."

Neanderthal Physique

Fossil records indicate that the Neanderthal were physically powerful and war-scarred race substantially more muscular than Homo Sapiens. They had extremely muscular chests, longer bodies, but shorter legs. Some fossil specimens , have hip sockets more splayed than modern humans.   A Neanderthal, partially because of this greater muscle mass, would weigh about 20-25 lbs [10-12 kilos] more than a modern human of the same height.  Physically, they were a race resembling the Incredible Hulk, with a greater intellectual capacity than their Homo-Sapien counterparts

One flaw in the Neanderthal/Nephilim theory is that Neanderthals, although of significantly greater body mass were actually slightly shorter in height than Homo sapiens and some claim they were unlikely to have been referred to as giants. This theoretical flaw is easily reconciled when one realizes that the word Nephilim actually means "fallen ones" and "Unnaturally begotten men"  not giants, the word being translated in the King James Bible as "Giants", is recognized now as a probable mistranslation.   

"Neanderthals although shorter than Modern man, and slightly shorter than their Homo Sapien contemporaries they had a much greater Body Mass ...“Neandertals were about 30% larger .than an average modern man and of great muscular strength. They had low foreheads, protruding brows, and large noses with broad nostrils and were meat eaters” ^{[1 ]}

."One of the most characteristic features of the Neanderthals is the exaggerated massiveness of their trunk and limb bones. All of the preserved bones suggest a strength seldom attained by modern humans."   -Erik Trinkaus Hard Times among the Neanderthals

"evidence indicates that Neanderthals were people of incredible power and strength—far superior to all but the most avid bodybuilders of today. "  - Bones of Contention: A Creationist Assessment of Human Fossils Marvin Lubenow

" Not only were they [Neanderthal] physically more robust than the gracile H. sapiens, they also had a larger brain, and there is some physiological evidence that they had superior sight, hearing and sense of smell."  Bryant Mansfield

Manual Dexterity / Opposing Thumbs

Anthropologists at one time theorized that Neanderthal was incapable of making the more advanced tools utilized by Humans because they lacked opposing thumbs and hence the requisite dexterity . Recent studies suggest otherwise.

"...some anthropologists have proposed that Neandertals died out because their clumsy hands were unable to manufacture advanced tools.  A new three-dimensional computer simulation of their thumb and forefinger, however, indicates that Neandertals had the same ‘precision grip’ as modern humans. 

To make their model, researchers scanned images of a thumb and index finger of fossils found in La Ferrassie, France, and then converted the images into a full-motion model (much like animators now do with computer graphics).  Although Neandertal hands look different from ‘modern’ hands (they had more muscle tissue and broader fingertips), their thumb was easily able to touch the tip of their index finger—the key to human dexterity."  Thumbs up for Neandertals
 

Neanderthal Language

Although no proof of a written language has been discovered. Neanderthal may have possessed an advanced vocalized language. Fossils indicate that their voice boxes contained vocal cords that were capable of making sounds similar if not the same as modern humans.
 

Neanderthal Communities and Culture

Neanderthals were a social race. The old and the injured were taken care of ,not simply left to die. Elderly Individuals have been found substantiating the belief that they exercised "humanity" in their exercise of mental thought principles over physical actions.

There is evidence indicating that Neanderthals possessed a sense of family / community spirit . Ritual burials and caring for the old and sick. Remains of physically impaired individuals have been discovered. Some skeletons display healed injuries which tends to indicate that the injuries were treated and cared for.

In a cave in Iraq, archaeologists uncovered skeletal remains of a man, two women and a child buried together in soil containing flower pollen .  The flowers, apparently being set there by survivors, strongly suggests that Neanderthals thought and cared for their dead.

In Yugoslavia a flute has been unearthed among Neanderthal remains which indicates that Neanderthals had the ability to make music as well being indicative of an advanced thought process.

 

Neanderthal Warriors

There is evidence that Neanderthals were vicious and warlike , suffering and inflicting battle wounds, mutilations, torture and maiming. One Neanderthal site containing human/homo sapien bone fragments shows signs of cuts and burns. These burnt and intentionally broken human bones have been interpreted as being evidence of cannibalism. They could also be indicative of torture and or mutilation by dismemberment as has often been the case within certain homo sapien cultures who engage in vicious battle blood lust.

Neanderthal DNA

Modern man and Neanderthal are both derived from a common ancestor who lived about 500,000 years ago, but  modern humans probably do not have Neanderthal ancestors in their family tree, a new DNA study concludes. The DNA extracted from the ribs of a Neanderthal infant buried in southern Russia 29,000 years ago was found to be too distinct from modern human DNA to be related. Not all Researchers agree. A 24,500-year-old skeleton found in Portugal demonstrates that Neanderthal man and early humans intermixed and produced children, said Erik Trinkaus, Ph.D., professor of anthropology in Arts and Sciences.

"It is believed by some people that modern man shared several thousand years of history with Neanderthals, and also that the Middle-Eastern region was home to some of the last surviving pockets of Neanderthals or H. neandertalensis. Therefore, it is conceivable that a folk memory of these creatures survived by way of mythology. In addition, it appears that the very last Neanderthals adopted some of the technological and cultural innovations of their H. sapiens contemporaries. The theory is that surviving Neanderthals or hybrids might have been large, powerful men possessing the intellect and societal characteristics of our own species, which would explain their identification as "mightiest ones" and "men of renown."  -Wikipedia