Informal proceedings are less complicated and generally take less time to complete. The results, however, can always be challenged in a formal proceeding. Thus, you do not have the finality gained by a court order obtained through formal proceedings. Formal proceedings require a court hearing. Notice of that hearing must be served upon interested persons.
Formal proceedings are more complicated and generally take longer.
They do give you a greater degree of finality for matters covered by the order. Supervised administration can be described as a series of formal proceedings from beginning to end. Generally, if there is no dispute over the admission of the will and appointment of a personal representative I would begin with informal proceedings. If it is likely that the will may be contested at some point or the appointment of the personal representative contested, I would consider using formal proceedings. There would seem to be very little advantage for requesting supervised administration since you can obtain court orders when necessary by using formal proceedings for a particular issue.