The word "conduzir" is more common and formal, whereas "dirigir" is often reserved for directing people or organizations.
It can mean supervising or "directing" a research project or PhD thesis. Dirigir
It is used for conducting an orchestra (e.g., a maestro directing a symphony) or directing a film or play. The word "conduzir" is more common and formal,
It is the standard word for "to drive" a car. Dirigir