Catastrophic APS can cause skin necrosis, but it is a systemic thrombotic disorder, not a primary, rapidly spreading soft tissue infection.
This causes intensely painful, retiform purpura and skin necrosis due to vessel calcification in renal failure patients, a different underlying pathology than an acute infection.
Simple cellulitis is a superficial infection; necrotizing fasciitis is distinguished by pain that is excruciatingly out of proportion to the visible skin changes, woody induration, and rapid progression.
Ecthyma is a localized, "punched-out" ulcer, not a rapidly spreading, deep infection of the fascial planes.
This is a syndrome of DIC and skin necrosis, often caused by sepsis (including from the organisms that cause nec fasc), but it is a systemic coagulation failure, not just a localized tissue infection.
Necrotizing fasciitis is a cause of sepsis; it is distinguished from other causes of septic shock by the presence of a rapidly progressing, incredibly painful, deep soft tissue infection.