Informal and formal proceedings are two ways to accomplish the same things. By using either, you can have a will admitted to probate and a personal representative appointed. 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.