The authority exercised is a kind of legitimate power and people follow figures exercising it, because their positions demand so irrespective of the person holding the position. Leaders in organizations and elsewhere may have formal authorities but they mostly rely on the informal authority that they exercise on people to influence them. Leaders are trusted for their judgment and respected for their expertise, integrity etc and hence followed and not because they hold a certain position. For e.g. M.K. Gandhi for most part did not hold any official position to lead the Indian freedom struggle.
It is also important to understand that a formal authority and power emerging from it, might not always be able to influence people in the desired manner as; in times of crisis and difficulties people view it as coercion. On the other hand leadership tends to create followers out of free will and choice without forcing them to accept anything thrown their way. Authority rarely provides a scope for feedback, constructive criticism or opinions of the people on whom it is exercised however leaders provide ample platform to their followers to voice their thoughts and feedback.
When dealing with adults, the sole use of authority to direct and discipline them hardly works, leadership provides a better approach of sharing and involving thus building rapports with followers and creating long term relationships. Authority can hardly make people change their attitudes and behaviors with lasting effects and results however a leader inspires followers through self modeled ways and hence leadership displays greater effectiveness in addressing attitudes and behaviors of people.
Exercising authority sometimes limits the approaches to arrive at solutions for issues and problems while leadership encourages people to look beyond the obvious and think innovatively and sometimes emerge with radical solutions.
Apart from it, the biggest difference between the two as cited by Stephen R Covey is the moral authority held by leaders over the followers which is absent in the case of power from authority. Within the organizational setup when leaders also have moral authority on their subordinates by establishing a synchrony in their words and actions; the rest of the structure and processes of the organization also get aligned to it, thus creating a robust and transparent culture.
Authoritative way of working also encourages individuals to work in silos while in the organizations of today; the leaders need to have a complete picture and coordinate with other functions and departments as and when required. It is indeed difficult for mangers and leaders to move out of their circle of authority and coordinate and interact with external people. However the need of the hour and the more effective approach to leadership and management is when leaders come out of their comfort zone and move from exercising authority on a small group to leading the entire organization.
Individuals, who do not rely on authority but lead people, are the ones who enjoy the privilege of their ideologies and thoughts practiced by later generations long after they are gone. Even with individuals who held positions of responsibilities, the ones who actually led their people are the ones remembered and followed.