The most basic distinction between a supervisory writ and an appeal is the appellate court’s discretion to grant or deny. La. C.C.P. art. 2201. See Livingston Downs Racing Ass’n, Inc. v. La. State Racing Com’n, 1996-1215, p. 3 (La. App. 4 Cir. 6/5/96), 675 So.2d 1214, 1216 (The difference between supervisory jurisdiction and appellate jurisdiction is that the former is discretionary on the part of the appellate court while the latter may be invoked by the litigant as a matter of right).
And while there are exceptions, supervisory writs are generally invoked for interlocutory rulings, while an appeal is the proper mechanism for review of a final judgment.
Appellate courts have both appellate jurisdiction and supervisory jurisdiction over cases which arise within its circuit under La. Const. art. V, §10(A)(A court of appeal has supervisory jurisdiction over cases which arise within its circuit). Meanwhile, the Louisiana Supreme Court has general supervisory jurisdiction over all other courts, La. Const. art. V, §5(A), and appellate jurisdiction under La. Const. art. V, §5(D).