I have struggled with this one when trying to explain it and have ultimately decided that "led" and "run" are too similar and interchangeable a pair of terms to try to hang this difference on. Instead I use "led" and "directed"
The leader is the publicly visible face of the unit. He plans and conducts events. He manages the discipline of others in the unit. He supervises sub-leaders or individuals in the unit. He sees to the training of new and continuing members of the unit. He controls the day-to-day operation of the unit by making the many little decisions needed to keep the unit running.
In scouts troop meetings, patrol meetings, campouts, outings, events, PLCs, Courts of Honor are all functions of the leaders.
The director is the behind the scenes controller of the unit. He sets the "big picture" goals and directions of the unit. He examines the operation of the unit to see that it is running as desired. He monitors the finances of the unit to ensure it will have the resources needed and, if not, takes action to increase income or decrease expenses. He controls the year-to-year operation of the unit by making the big decisions that create the identity and philosophy of the unit.
In scouts Committee meetings, SM conferences, Boards of Review, Committee Meetings, and finances are all functions of the director
A troop is youth-led, adult-directed.
A crew is youth-led, mostly youth-directed.
An in either case there is an adult executive to make sure it doesn't run off the rails.
