According to the regs:
”Authorized instructor” - Instructor that holds FAA teaching credentials.
Part 61 - I can teach at a picnic table under a shade tree. Not required to use an “approved syllabus”.
i can use any and all available syllabus or make my own.
Part 141 - “FAA approved pilot school” needs inspected and approved classrooms, teaching aids, air conditioning, rest room facilities, weather briefing options (online or phone) and a syllabus which is approved for use by the supervising FAA inspector. This can be a commercially available or self made syllabus but it needs to be “approved”.
These schools are subject to stricter FAA oversight, record keeping requirements, record checks and flight instructor standardization requirements.
In an extreme case ( just to clarify the point) a part 61 school can employ a dozen Part time flight instructors who each use their own chosen syllabus to the best of their knowledge and ability.
Part 141 school - all instructors have received standardization training and use the same syllabus and are subject to stricter record keeping requirements and pass rates.
The school I worked for did not differentiate between Part 61 and Part 141 and treated ( and trained) everybody to Part 141 requirements.
Consider a Part 141 flightschool like a chain restaurant. They all offer the same menu and answer to corporate.
Part 61 can be ( not always is) like a local mom and pop restaurant.
Only Part 141 schools can train non-US students on a student visa.
Not counting green card holders.