One has a plural noun, the other singular.
There seem to be several idiom pairs/related idioms with inconsistent grammatical number. I am also thinking "on the fringe" and "on the fringes."
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
They are two different idioms, the latter being formed from a situation in which one is unable to grasp something, even with both hands.
The difference between the two:
I think it has to do with the original idea of "hand" in the sense of "control" which contrast with "in hand" under control. In that respect, the usage of hand in the singular form is metaphoric and idiomatic:
- Out of hand (1590s) is opposite of in hand "under control" (Etymonline)
From The Phrase Finder:
-Out of hand -- "If you reject an offer or idea 'out of hand,' you do so without hesitation. However, this phrase has several different meanings, the oldest of them being 'out of control,' from the days when failure to keep a firm grip on the reins would result in a team of horses being 'out of hand.'" From "Morris Dictionary of Word and Phrase Origins" by William and Mary Morris (HarperCollins, New York, 1977, 1988). Page 433.
The "out of hand = out of control or not being processed" meaning contrasts nicely with the "in hand = under control or being processed" meaning.