Classical management theory includes the scientific management theory of Frederick W. Taylor, who pioneered the idea that each job should be studied carefully to ascertain its requisite components, as well as the skills and training necessary to complete each task, and that the gathering of this information should be used to create a quota, a minimum that each employee should be expected to produce. Otherwise, he thought, people would drag out tasks as much as possible, since their only motivation was a paycheck.
Weber is also considered as part of the classical management school of thought. He theorized that a hierarchy in a business, with a stable cadre of administrators who ran the company, would provide a more stable and productive business environment. This was the origin of the bureaucracy, a term we use with derision today, but a concept that still has great utility. In government, the civil service employees comprise this cadre.
Still a third aspect of this management school of thought was what is referred to as "time and motion" studies. Frank and Lillian Gilbreth, who are the parents in the original film version of Cheaper by the Dozen, a mostly true story, based upon a book by their son, pioneered time and motion studies. They observed countless tasks and ascertained how those tasks could be accomplished with less wasted time and motion. Interestingly, they tried their ideas out on their twelve children.
Behaviorism, to be sure, concerns itself with the behavior of individuals, but it posits that behavior can be controlled through reward and punishment, reward being called positive reinforcement and punishment being called negative reinforcement. Management theory has gone beyond this for the most part in contemporary practices, focusing rather on what some call humanistic management theory, acknowledging that in today's mostly post-industrial world, our expectations of employees are not to produce x number of widgets per hour, but to be creative, think critically, take the initiative, and solve problems. Behaviorism is not all that effective at motivating employees to succeed in this milieu.
The main difference is the human factor.
Classic management theories were studied throughout the first part of the 20th century in aims of trying to create a one-step model of understanding the trends and patterns of an organization. The aim was to create a system of prediction of actions, dynamics and interactions within a business, as a whole.
However, as different individuals became employed from all sort of backgrounds and age groups, different situations began to arise; after all, an organization is the sum of all its components and changes as they also change. For this reason, a behavioral approach was taken to treat management issues that dealt with people in a people to people basis. Hence, the behavioral school was able to differentiate individuals by their backgrounds, walks of life, age groups, ethnicity, etc to understand their unique needs.
Keep in mind that this paradigm continues as we do not want to confuse "understanding" diversity with "categorizing" or "labeling" it. Therefore, the behavioral school aims to continue the research on interactivity and dynamics within organizations with an emphasis on human nature and personality traits.