The devil finds work for idle hands
This proverb, which has been traced back to Geoffrey Chaucer's The Tale of Meilbee in the 14th century, suggests that boredom often leads to acts of mischief that do more harm than good. It's gone through many iterations in the last several hundred years:
From The Tale of Melibee, c. 1386:
Therefore seith Seint Jerome: "Dooth somme goode dedes that the devel, which is oure enemy, ne fynde yow nat unocupied."
I. Watts, Divine Songs for Children, 1715:
In Works of Labour or of Skill I would be busy too: For Satan finds some mischief still for idle Hands to do.
From T. Fuller in Gnomologia, 1732:
Idle Brains are the Devil's Workhouses.
And from H.G. Bohn's Hand-Book of Proverbs, 1855:
An idle brain is the devil's workshop.
These last two versions are closest to the one I heard while growing up: "idle hands are the devil's playground." My grandparents and teachers would trot this one out whenever my brother and I started to get fidgety and whine that there was nothing to do—and then they'd rope us into raking leaves or weeding the yard or, if we were at school, they'd give us extra work to do. To keep us "out of trouble," they said. I mean, you let your brother climb onto the roof to rescue a Frisbee just one time, and you never hear the end of it!
And that's the point of this proverb: when we're bored, we tend to get ourselves into trouble by doing ill-advised things that seem like a good way to inject a little excitement into our lives.
The foot-high stack of books sent up a billowing cloud of dust as Bobby dropped them onto the desk. Dean coughed, scowling. "What's all this for?" he asked.
"The devil finds work for idle hands, boy. If you don't keep busy with useful research, you're bound to go haring off all half-cocked and try to find a way to rescue Sam that screws him up even more," Bobby said. "And then Lucifer's gonna be that much further along in his plans."
The idea here is that boredom or a lack of anything useful to do results in thoughtless acts that end up doing damage, or "helping the devil"—literally, in Dean's case. So it's better, the proverb implies, to keep yourself occupied with useful, helpful things lest the devil use your idleness to further his own ends. In more secular terms, the devil finds work for idle hands just means that when we're bored, we're more likely to do something stupid and make trouble for ourselves or those around us.
Many hands make light work
First recorded (in the English language, at least) in the knightly romance Sir Bevis of Hampton, this proverb from the early 14th century simply means that the more help you have, the easier a task becomes. This idea was apparently a very popular one because the saying has appeared in nearly all proverb collections since 1546, and it works very nicely in fanfiction, too.
Sam glared up at Dean from the half-empty grave. "You could pitch in, you know." He jabbed his shovel into the ground. "I mean, they say 'many hands make light work.' We'd be done in half the time."
Dean grinned. "Grave's too small. Anyway, they also say 'too many cooks spoil the broth.'"
"Hey, Cas," Dean said in surprise as Castiel stepped into the room. "We weren't expecting you until tomorrow."
"My brothers joined the battle. Many hands make light work," Castiel replied.
Dean snorted. "Yeah, especially when those hands are angelic."
Your characters would probably use this saying when they're trying to get someone to help out, like Sam in the first example above, or when they're observing that the help they've received has made their task easier, as in the second example. Because that's the whole point: a pair (or three) of helping hands gets work done faster and easier than doing something on your own.
So what's the connection between the two proverbs? The idea of being gainfully employed in some manner, either to keep yourself from inadvertently causing some mischief for you or others, or to share the workload with someone else to make a heavy task lighter. In both cases, we get the idea that keeping busy is better than the alternative—as I'm sure the Supernatural boys, my grandparents, and my former teachers would all agree.
Idle hands are the devil's tools. (The Phrase Finder)
Many hands make light work. (yourdictionary.com)