Beyond the fact that they mention children, they have very little in common.
Not in Front of the Kid is when characters act 'better' than they normally do because a kid is present (though people other than kids can fill that role).
Think of the Children! is about Moral Guardian types claiming that X will somehow cause harm to children and that X should be banned or at least heavily restricted because of that. Generally there is little to no evidence that X will actually harm kids.
So Not in Front of the Kid is if it's done In-Universe, right?
We don't need justice when we can forgive. We don't need tolerance when we can love.
Umm.... no.
Not in Front of the Kid would be something like "I swear like a drunken sailor most of the time, but I don't swear at all if there is a kid around"
Think of the Children! is "Short skirts are causing the moral decay of our youth!!! We must force women to wear ankle length skirts/dresses or else our precious children will all grow up to be murderous drug addicts!"
There is no In-Universe distinction, there is simply no meaningful connection between the two.
What's the difference of Think of the Children! and Not in Front of the Kid?
We don't need justice when we can forgive. We don't need tolerance when we can love.