Acute procedures are emergency and life threatening situations such as a car crash, heart attack, flu epidemic or broken leg. Elective procedures, while often critical, are procedures the patient chooses to have such as prostate/bowel/breast cancer surgery & treatment, hip replacement, heart bypass or varicose vein surgery.
A private hospital doesn’t have an emergency or intensive ward so is not equipped to deal with acute procedures while a public hospital may become so overloaded with acute situations they are unable to cope with elective procedures.