What I mean is, when you treat someone "like an adult", you let them be responsible for their own decisions, you let them suffer the ill consequences of their own mistakes, and you listen to their words as if they mean what they say literally. When you treat someone "like a child" , you clean up after them, you fix their mistakes for them, you assume that when they say something wrong they didn't mean it, etc.
I think some examples may clarify what I mean.
Say you're going hiking in the mountains with a friend. You notice that they have not brought a jacket and you know it will be cold up there. You say "hey do you want to borrow a jacket?" and they say "nah, I'll be fine". You know fully well they will not be fine. If you "treat them like an adult", you would just let them suffer the ill consequences of their bad decision, but the result will be unpleasant for you as well, they will complain, they'll be all pouty and in a bad mood, they'll want to leave quickly, it will suck. Either you can say "fuck you, I told you to bring a jacket, I want to stay, suck it up!" or you can accomodate them and leave early, and either way sucks. So the better thing is to "treat them like a child" and just say at the start "well I'll bring an extra one anyway in case you want it". (with particularly childish friends you shouldn't even say anything and just silently bring an extra one).
(The same goes with snacks and water and such obviously; basically you're better off being like a mom and carrying a pouch of supplies to keep all the "children" (ie. all humans) from getting cranky).
Say you're driving with your dad and you're lost and he doesn't want to stop for directions. If you treat him "like an adult" you would either just speak to him rationally and say "hey this is silly you need to stop and ask someone, don't be so childish" or you would just let him suffer the ill consequences of being lost. But of course neither of those would actually work (almost nobody responds well to having their bad behavior pointed out to them). What you need to do is treat him like a pouty child and fix the situation yourself; eg. say you really need to pee, can we stop for that please, and then ask for directions yourself.
A very common one is just when someone is really pouty or starts acting like a jerk to you. You could "treat them like an adult" and assume they are aware of what they are saying and actually mean to be a jerk to you. But in reality they probably don't, they are just hungry or cranky or need a poop (they are a child) and you shouldn't take it personally. If you need to interact with them, you should get them some food and water and try to fix their crankiness before proceeding.
I find in general that interactions with people work much better if I treat them like a child. (and the same goes in reverse - I get along much better with people who treat me like a child). (basically the idea of the rational self-responsible adult is an invention that does not correspond to reality)
(I guess a related thing that everyone in "communication" knows is you can't just criticize someone and expect them to rationally accept the information and decide if it is useful or not; you have to butter them up first and do it really gently and all that, just like you were trying to critique your child's drawing ("that tree is awful, trees don't look like lollipops, you moron"))
(I guess 99% of modern publicity is just treating people like children. It doesn't matter how good your product is if it has enough stars on the box; your store can sell garbage if it smells like cookies; it doesn't matter what the president actually says as long as he has good hair. I feel like in the 50's before PR was figured out, that media actually treated adults like adults a bit more, and the cleverness of the modern age is realizing that everyone is an easily manipulated pouty child (suck on your iNipple))
Related : thoughts on using money.
I have enough money now to live comfortably, much more so than when I was a child. The little differences are really what strike me. When I was a child of course you would buy store brand aluminum foil, of course you would use coupons, those dollars all mattered. Buying food at Disneyland was a huge luxury (to be avoided if possible, you can wait till we get out of the park, right?) because it was marked up so much. So the first good use of money is just hey you can buy whatever basic necessities you want and not waste your time worrying about the price.
I've tried various ways of spending money now and think that I've made some discoveries. Fancy cars and fancy houses are not good ways to spend money. They are not any better and don't improve your life. In general buying stuff/goods/toys is not helpful (except when it allows you to do an activity that you could not have otherwise done, and you actually do that activity and enjoy it; eg. buying fancy road bikes has zero value if you already had a bike that was good enough and you enjoyed riding; if it doesn't change your ability to do an activity, just making it faster or easier or whatever has zero actual value; but if you had no bike and buy one and then actually ride it, okay that's a good use of money).
Anyway, one of the best uses of money is just to fix all those little moments of crankiness. Like you're in a museum and you're kind of tired or hungry or thirsty; you start to get cranky and not enjoy it. My depression-era upbringing tells me to just gut it out; stay the hell away from the museum cafe, because it's crap food and it's way overpriced. But with money you can just buy the ten dollar tuna sandwich and it will fix your bad mood; that's a good use of money. (in my youth we would have brought homemade sandwiches).