The way I understand it referenced to a large aircraft:
121: public charter: supplemental (non-scheduled airline)
125: private use: large aircraft version of 91, and can be negotiated with the FAA to combine parts of 91 and 125 depending on the stated mission for the aircraft. This was the simplified version as told to me by the FAA when I was involved in determining requirements for an appropriate operating certificate for a company who was considering purchase of a large aircraft for research purposes.
135: smaller aircraft only, but I have seen Twin Otters operated under 121 (scheduled ops)
The best course is to have a face to face chat with the prospective FAA POI.