When someone is breathless, he is unable to get enough air into their lungs. Breathless can also be used to describe someone who is holding her breath due to fear, suspense, or astonishment.
Ex: Hannah was breathless after climbing up ten flights of stairs with her groceries. Normally she takes the elevator up to her apartment, but it was out of order.
Ex: The audience was breathless as the circus performer carried out an amazing feat. Once it was clear that the circus performer would not cause himself harm, a collective sigh of relief washed over the audience.
When someone is short of breath, they are panting or having trouble breathing. Breathless and short of breath can usually used interchangeably, but short of breath is typically used in medical contexts and when the difficulty breathing seems out of proportion for the physical activity that the person was doing before losing his breath. The condition of having trouble breathing is known as shortness of breath.
Ex: Sergio went to see his doctor because lately he has been experiencing shortness of breath. His doctor ordered tests to make sure he isn't suffering from heart disease.
Ex: It is important to never ignore feeling short of breath. Breathing difficulties are a symptom of many different illnesses, some of them very serious.