From the past two weeks while I was "otherwise occupied" with work and recuperating, I have such a backlog of DD words, that I think I can keep up the French trend for some time.
Moil derives from the Latin "mollis" for "soft" as related to "mollia (panis)" for "the soft part (of bread)", through Old French "moillier" for "to soften as from having been made wet" and eventually to Middle English "moillen" for "to soak or make wet" which of course led to manual labor because why else would anyone c. 1400 get wet other than toiling in the mire??? And that's the definition that we are left with--toiling, drudgery and hard work, and some latent description of the churning water as from the labor in the mire. Ironically, this word has no relationship to turmoil, which derives from some mill activity.
Now, compare toil, which derives from the Latin "tudiculare" for "to stir up or beat" as from a machine which crushed olives, through the French "toiler" for "to contend". It is hard to see how "crushing" becomes "contending", but ultimately, in Middle English c. 1250-1300, the word came to mean just hard and continuous labor. Maybe they were just contending against the daily grind. :-)
So, how does moil differ from toil, other than the initial consonant? Well, since moil derives from water, there is still the residual implication of churning in the work, while toil is has the residual crushing. Therefore, the busy receptionist's attention diverted by walk-ins, phone calls and doctor requests added to the moil which was her work environment, while the incessant HMO paperwork only added to her toil. The first is a flurry of activity like a whirlwind, while the second is the crushing blow. Daily teacher's toil away under adverse circumstances of dwindling resources, striving to keep each classroom from creating a moil. I like the first usage, but the second becomes a bit vague. Children after eating too many sweets are a moil of restless energy? Better, but still a little opaque. Depression moils in a downward spiral unless checked. eh. Basically, I don't like this word, even if it does have a French etymology. It's a bit useless and obscure. I'll keep working at it. See if I can make it work for me without too much moil--or toil.