Part 125 covers commercial ops of large aircraft that don't hold out or do common carriage. If you've got more than 20 seats or 6000 pounds payload, you operate under 125 v. 91. VIP 727s and sports team aircraft operate 125. There have been some 125 cargo operations, Florida Air Leasing just hauled Wall Street Journals and some operators in Mich only worked for the Big Three auto makers.

Gets you out of all sorts of flight and duty time limits and training requirements, only need a COMM as PIC. Oddly enough, 125 does not get the same extension of the F/E written that 121 does.

And both the reg and the FSIMS just say you have to be employed by a 135 air carrier, nothing about scheduled v. non-scheduled.