As in this WWII poster:
Are they the same thing, or are there differences in expression？
Why do native speakers choose the first one?
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You're slightly misquoting. The famous poster said "Hitler will send no warning" (singular), and referred to a hypothetical use of chemical weapons on British civilian targets, which it turned out didn't happen.* That is to say, it refers to a single attack; the first attack that could happen.
Alternatives therefore would also include "Hitler won't send a warning" or "Hitler won't send any warning".
That point aside, you are correct in deducing they are equivalent. It's also more common to use a negative form of sentence, rather that "no" before the object.
The difference is stylistic, not one of meaning. It puts the emphasis on "no warning", and hence highlights the risk civilians were at, and by extension, the importance of carrying your mask, and practising your gas-mask drill. (The trick is to point your chin toward the sky, and let it "fall" onto your face, and you should have a secure fit in less than 3 seconds).
*Partly because chemical weapons would have been a war-crime under Article 171 of the Versailles Peace Treaty, and partly because it was clear to both the Allies and the Axis powers that the danger of retaliatory use of chemical weapons was worse than the potential gains. The Japanese did use such weapons, but only against non-Western opponents. The Germans used chemicals in the massacres of civilians in occupied Europe during the holocaust, but not against military opponents. The only deaths from chemical weapons in the War in Europe were caused by the accidental dispersal of American mustard gas (that they were planning to use in retaliation, had Germany resorted to such weapons), after supplies were bombed, killing America seamen, and Italian civilians.
"Hitler will give no warning" has the same meaning as "Hitler won't give any warning". There is no difference.
In everyday speech, we are more likely to put the 'no' before the object to show absolute certainty as well as to emphasise the importance of the message. It tells the listener that the issue is not up for debate.
Consider: "There is no excuse for this behaviour" Vs "There aren't any excuses for this behaviour".
The use of the word "no" shows you are gravely serious.