hedyot and idiot

What is the connection between the Hebrew hedyot הֶדְיוֹט - "layman, layperson" and the English "idiot"?

They share a common origin, but in this case the Hebrew is closer to the original meaning than the English is. 

Hedyot entered Hebrew in the rabbinic period, being borrowed from Greek. It was used in phrases like כֹּהֵן הֶדְיוֹט kohen hedyot (as distinguished from the High Priest), or in this mishna, discussing permitted work on the intermediate days of the festivals:

הַהֶדְיוֹט תּוֹפֵר כְּדַרְכּוֹ, וְהָאֻמָּן מַכְלִיב

A layman, who is not a skilled tailor, may sew in his usual manner if necessary for the Festival, whereas a craftsman may form only temporary stitches. (Moed Katan 1:8)

Klein notes that its original meaning of hedyot was "a private man, a layman, a common person" and provides this etymology:

Gk. idiotes (= private person, one not holding office; layman; an ignorant), from idios (= one’s own, private, personal, separated, distinct)

The same idios gave us two other English words that preserve that original sense of "separate, private":

• idiom: "phrase or expression peculiar to a language"
• idiosyncrasy: behavior or thought particular to an individual

But what about "idiot"? That seems to have a different meaning than the words we've mentioned so far. Well, it took a longer journey:

early 14c., "person so mentally deficient as to be incapable of ordinary reasoning;" also in Middle English "simple man, uneducated person, layman" (late 14c.), from Old French idiote "uneducated or ignorant person" (12c.), from Latin idiota "ordinary person, layman; outsider," in Late Latin "uneducated or ignorant person," from Greek idiotes "layman, person lacking professional skill" (opposed to writer, soldier, skilled workman), literally "private person" (as opposed to one taking part in public affairs), used patronizingly for "ignorant person," from idios "one's own"

So like hedyot, it originally meant a private person (as opposed to a ruler like a king) or a common soldier (as opposed to a military officer). Only later did it come to mean someone uneducated, then a more "technical" term for someone mentally deficient, and eventually in our day simply an informal term for a foolish or stupid person. 

That last meaning exists in modern Hebrew as well - as אִידְיוֹט (borrowed from European languages). Despite their similarities, it's important for any speaker of Hebrew today to not use them interchangeably.