Neosporin with vs without bacitracin zinc, what's the difference?

NONE. Both worthless. Application to wounds, burns, surgery, or anything does not prevent or cure infections. They are among the most common agents that people are allergic to. They cross-react, so allergy to one often means allergy to the other. They often interfere with wound healing. Their use contributes to emerging antibiotic- resistant bacteria. Each has been "allergen of the year" in different ye.