I can't think of a situation where you couldn't use one or the other She left with no remorse She left without any remorse without + verb is more standard She left without saying goodbye

The above is correct but if you were to say something like 'she left without him', it would not work; 'she left with no him' does not make sense. Generally, just do not use 'with no' when talking about people, or specific animals (e.g. 'she left without her pet' =/= 'she left with no pet'. Both sentences means she left without an animal however the first tells us that she left without *her pet*. The latter implies that she left without *anyone's pet*)

I agree with J. Rios. His examples are good and generally they mean exactly the same thing. I note also that "without" can be used in some additional situations where you could not use "with no". The most common uses would be things like "without much", "without many", "without a lot", etc. In those situations you could not use "with no."

I agree with Fifi. You never would want to use with no when talking about a person. Also, in sentences like "With no amount of pressure could I persuade her" without would not work. "With no" is often used more when quantifying specific "things". You could say "I'm going with no other people" becuase you are talking about people as an object like you might say "I'm going with no extra money" but you would never want to say "I go with no Karen", you would need to say I go without Karen.