What's the difference?
She doesn't speak English.
She speaks no English.

Question

Answer

Both are correct and mean exactly the same thing.
"She doesn't speak English." would be read and heard much more often.
"She speaks no English" would probably only be found in written English and even then, not as often as the first form.

Answer

There is no difference. 'She doesn't speak English', is far more common. I rarely hear anyone say the second one.

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Some believe that the second sentence is more emphatic (and even a bit angry sometimes):

Joe: Why don't you ask Mona?
Tom: Because she does not speak any English.
Joe: Please ask her. 
Tom: Didn't you hear me? I said that she speaks NO English!