The confusion over the etymology of countenance is a direct result of mistranslations in Genesis 4:4-5 (KJV)
4And Abel, he also brought of the firstlings of his flock and of the fat thereof. And the LORD had respect unto Abel and to his offering: 5But unto Cain and to his offering he had not respect. And Cain was very wroth, and his countenance fell.
Respect was never correctly translated. At that time in history 'respect' meant 'payment in kind' from the Latin word 'specie' meaning 'coin.'
When Cain didn't receive respect (payment in kind of coins of equal value), his countenance fell.
As you can see from most of the previous answers there is no clear etymological connection between face and countenance other than the reference the KJV circa 16th century.
Obviously most of the other scholars ignored the real root of countenance (count) and tried to force some other meanings to match the mistranslations of KJV.
From Middle English counten, from Anglo-Norman conter, from Old French conter (“add up; tell a story”), from Latin computare, present active infinitive of computō.
You then add the suffix '-ance' meaning 'the action or process of doing something' to the Middle English 'counten' and you have created the word countenance from Genesis 4 KJV.
Our timeline for the word now works well: Middle English (1066-1500) Counten is enumerating and countenance is the process of enumerating, then the KJV is translated from 1604-1611 and that is when the first instance of counten or count or cont had anything to do with a face.
You can study the etymologies of any variation of count, cont, con, etc. and you will never find a connection to the word 'face' or anything like it.
There are other words mistranslated in the verses. There is no evidence of "fat" referencing subcutaneous tissue before Middle English. 'Derma' and its variants had been the accepted root for 'flesh' or 'skin' for quite a long time at that point. 'Fat' and 'Pat' had a unwavering connection to the words 'father' and "paternal" from at least 3500 bc.
If 'fat' is 'father' and 'thereof' means 'origin,' then, "Fat thereof" seems to mean "paternal origin." That makes a lot more sense in relation to the phrase "firstlings of the flock," than referencing the origin of its subcutaneous tissue.
When I consider that the first use of the word 'sheep' in reference to a lamb or a ewe was KJV, it leads me to other ideas about why the Genesis account (KJV) called Abel a "keeper of sheep" instead of "shepherd."
"Tiller of the ground" means "tiller or Taylor of the ground loom," and Cain's offering of the "fruit of the ground" would be "fruit of the loom." In keeping with the weaving theme, Cain was said to 'slay' Abel but 'slay' almost certainly didn't mean 'to kill' until the 14th century. At that time, and still today, slay, sley and sleigh were the reeds of a loom or the thread count of woven fabrics. (Another reference to weaving and counting)
Unfortunately, references to weaving in the primitive languages of that area are almost always mixed in with references to storytelling as weaving reality on a loom. It's hard to tell if Cain was a weaver or a storyteller, probably a bit of both, but he was most definitely not a farmer.