TLDR: If you're in extreme Arctic conditions fur does have an advantage.
There is this study on the Inuit's Fur Parkas
ABSTRACT: The traditional clothing system developed and used by the Inuit is the most effective cold weather clothing developed to date. One of the key elements used by the Inuit is a fur ruff attached to the hood, hem, and cuffs of their parkas. This paper determines why the fur ruff is so critical
to the effectiveness of cold-weather clothing, especially in protecting the face, without impeding movement or view, so essential to the Inuit hunter. To quantify the effectiveness of this clothing, heat transfer was measured on a model placed in a subsonic wind tunnel. The wind velocity and angle to
the wind were varied. A boundary layer forms on the face, and heat transfer was measured across that layer using thermocouples. Different fur ruff geometries were examined to determine which was most efficient. The experimental results were combined with data collected using ethno-historical methods since 1970 by 2 of the authors. The traditional headgear proved to be the most efficient. The lowest heat transfer was found for the sunburst fur ruff geometry at different angles of attack and wind speeds. This unique combination of scientific and traditional Aboriginal knowledge provides a holistic perspective on new insights into the effectiveness of cold-weather clothing systems.
The face must be exposed in order to see for hunting
or travel. It is therefore essential to minimize facial
heat transfer for frostbite prevention and survival. Not
only does the Inuit parka ruff, which is worn behind
the cheek bones, allow the wearer to see while hunting
and traveling, it also drastically reduces the amount of
hoar frost that otherwise builds up around a hood
if worn in front of the cheek bones (as is the case
with southern-style hoods). This frost build-up impairs
vision and acts as a conductor, drawing heat from the
face, thereby increasing the possibility of frostbite.
The superior effectiveness of this piece of clothing
has been known by Inuit hunters and seamstresses,
who have thrived for thousands of years by creating
polar ruff designs that provide protection against the
cold, windy arctic climates; it is quantified for the first
time in this paper. The sunburst fur ruff design is truly
a remarkable ‘time-tested’ design.
There is also what Canada Goose who makes fur coats says.
Why we choose fur.
No matter where they’re worn, many of our products are designed and built to protect against the elements in the coldest places on Earth – places where exposed skin can freeze in an instant. In these environments, we believe that fur is the best choice. Having fur trim around a jacket hood disrupts airflow and creates turbulent air which helps protect the face from frostbite.
In addition to a banquet, the AAC took the anniversary year to get some gear from the 1963 expedition in a studio for a photo shoot. This included a jacket made by Eddie Bauer that's among the most iconic in the history of outdoor sports.
Bright red and equipped with a fur ruff on the hood, the jacket was state of the art in its day. It was insulated with enough down to be protective to minus-85 degrees.
Whittaker kept the jacket packed away for most of the ascent in 1963.
It was too warm until the climbing team got high on the mountain's face.
From the NPS page on caribou skin clothing,
Inland mountain Eskimos experience one of the world’s most extreme winter climates—temperatures of 55 degrees below zero or colder, often with gale force winds and blinding snow. Despite these daunting conditions, Eskimo people carry on with their daily life of hunting, fishing, gathering firewood, traveling, and camping. The key to their success and survival—above all else—is warm, effective, brilliantly designed and expertly made clothing.
The Eskimo people make their warmest clothing from caribou hide—a material that evolved over millions of years in the Arctic environment, providing caribou with unequaled insulation against penetrating cold and gales. Caribou hair is hollow, so it traps insulating air not only between the hairs but also inside them. Clothing made from this material is extraordinarily warm, lightweight, water repellent and durable.
An Eskimo hunter dressed in traditional clothing was completely wrapped in caribou skins. His parka —a hooded jacket invented by Eskimos—was made of caribou skin and worn with the fur inside. For deep cold and storms, a second parka could be worn over the first, with the fur side out. A wolf or wolverine fur ruff around the hood created a little pool of warmth that protected the wearer’s exposed face. Unlike other furs, wolverine also easily sheds the frost that collects from a wearer’s breath.
Caribou skin boots (kamik) are durable, extraordinarily warm, and nearly as lightweight and supple as the most comfortable slippers. No modern materials can match the combination of warmth and light weight of caribou skin boots.
Caribou skin boots, socks, and mittens—meticulously crafted by women in the village—are still regularly worn by Nunamiut people and are regarded as superior to any commercially made substitute.
Perhaps the strongest testament to the ingenuity and effectiveness of traditional Eskimo clothing is seen in the iconic bright red parkas with fur ruffs, worn by Antarctic researchers working in the coldest places on earth.