Both verbs are used within the world of corporations, companies and enterprises.
However, there is a significant different in their usage in legal English language.
‘Manage’ refers to the day-to-day operation of a company. Managing a company includes hiring and managing employees, organizing daily tasks, etc. The successful management of a company requires, above all, good organizational skills.
The verb ‘direct’ refers to the direction of the whole company in a broader sense. Directing a company includes defining the overall vision of the company, its main objectives and priorities.
In the civil law world, management is used for activities performed within the company. A manager is part of the organization, who works together with other employees. THe director is often perceived as an external officer, often a member of the board of directors, whose role is more a leadership function rather than being engaged in day-to-day operations.