What are the main differences between Leadership and Management, and how should this be applied in terms of career advancement?

Leadership is a pure human ability, to open a path and have the ability to drive others into this path. A leader is more an inspiring person, who motivates others.

Management is the technique of analyze data, forecast needs and manipulate sources to achieve a goal.

This doesn't mean that a manager shouldn't be also a leader or the opposite. 

But it's different skills, the one more talent oriented and the other based on experience and education.

Leadership needed in Management/Manager  for success all around.

A Manager with Higher leadership qualities bag all success in career and career development if gifted with a success platform.

Management is procedural approach in order to maintain consistency of the end result. Yet, leadership is behavioral style that requires decent emotional inteligence so that the leader will be able to get the best out of his team. Management focuses on resluts while leadership focuses on inspiring & envisioning the employees so that they maintain a continuous momentum towards better results and not just achieving results.

As for career advancement, for sure leadership rules as succession planning and advancement will not happen just because maintain solid procedural system, it needs moral & synergy.