Shortness of breath is usually abbreviated to SoB - note the lower case 'o'. Generally, a patient will complain of being "short of breath", but will be described by staff as being "dyspnoeic", but as Dr Bill says, they're pretty much the same thing. Breathless(ness) is used whenever it sounds better than short of breath. The main difference is that consultants use "dyspnoea" and the junior medical staff use SoB.

Oh, and I've never heard of agonal breathing.