How “Devil may care” is different from “After me the deluge”?
Solution 1:
The phrase "devil-may-care" seems to refer to general carelessness, while "Apres moi le deluge" (originating with historical King of France Louis XV) seems to refer to a specific carelessness motivated by the fact that after the individual being careless is out of the position in which they have something to be careless about, the consequences have no effect on them. So, they can not exactly be used interchangeably, as someone who is being careless (someone who "is devil-may-care") might not be doing so because their thought process is that of "apres moi le deluge", but rather for some other reason.
Solution 2:
To add to @Joe's answer (sorry not enough rep to leave a comment), they are not the same, because 'devil may care' implies the person simply does not care, while the quote from the King of France implies the person is beyond caring (they cared once but no longer), or they are powerless to take care, because circumstances have overtaken them, having your head cut off being an extreme case of incapacity.
Solution 3:
To me, devil may care is so familiar as to have become effectively a word, while after me the deluge is a literary French phrase that I am not sure I have ever encountered in English before. It is certainly not an idiom in any kind of English I am familiar with.
Solution 4:
"Devil-may-care" by being paired so frequently with "attitude" has taken on an implication of "seeming" to not care, an affectation of not caring, a pretense. "When the Devil make hair, devil may care." -Vogue (Usage of the royal "We" to conjugate make.) Quote refers to men's fashion for wearing tussled hair (bedhead, windblown) being a gallant feat of "fête galante" effete. We may care after what We choose, for We are the Devil by choosing. Choice is chaos and chosen chaos divines better beauty by design. In this example, "devil-may-care" trues right out of what would otherwise be left a lying mess.
"Après moi le déluge," doesn't leave the devil standing to "may-care" or not. Let the waters rise or set the world afire to burn away until its last ash consumes. Come what may, for the triumvirate of God, the Devil, and I will already be well removed unto oblivion. "Après moi le déluge" is literally a point when no consequence, no matter how great, can be brought to bear. After the flood, the world's gone, the devil's dead, and there'll be no hell to pay.
When King Louis XV supposedly said, "Mon Dieux, Après moi le déluge," he was saying in no uncertain terms that His Majesty was all that stood between the government and annihilation as catholic as the Great Flood. And, he was right. After his death, came the French Revolution. Within 15 years, a systematic genocide visited upon the French Royal Family by the revolutionaries delivered the French Monarchy into extinction. Anyone anywhere in the world with even a notion of a claim to the throne of France was hunted down, dragged back to Paris, separated from their heads, and then left on display for all to bear witness, leaving no room for error or doubt, that no one was left on the face of the Earth that could ever inherit France again. On this note, the French bested God Himself, for even The Bible accounts for a decimate who survived after the Flood, aside from Noah and his family. However, the tide of France did not tithe, neither a livre nor a life had as lief dyed heart's sleave afore head's leave left dead at friends and no hope of home alive.