Let me first explain what formal and informal authority mean.
Formal authority is not gained, and the person does not have to work in order to earn it. Formal authority is granted based on the person's position and according to the organizational hierarchy followed. For example, in typical functional organizations, the IT manager has formal authority on all the resources in the IT department, who report directly to him, and must prioritize his tasks over anyone else's.
A perfect example of an informal authority is that of the project manager. In non-projectized organizations, Project Managers do not have direct authority over the resources, and thus have to gain this authority by gaining the resources' respect through their (the project managers') proven skills and leadership. Informal authority is hard to earn, as most resources know that the Project Manager do not have any authority over them, and sometimes their direct functional manager does not help in that area. Functional managers do not like the idea of someone else having authority over their resources.
Formal and informal authority apply across all organizational types (functional/matrix/projectized), although in the latter one, the Project Manager does possess some formal authority over the resources.
To conclude, here are, in short, the differences:
Formal Authority is:
- Granted by the hierarchy of the company to the position
- Is easy to earn (provided the person has the position)
- Can be easily enforced
- Is protected by the company's culture, and thus very hard (sometimes impossible) to break (this is why everyone tells you to avoid having conflicts with your manager)
Informal Authority is:
- Is hard to earn
- Can be easily taken away
- Does not enforce itself, instead, employees choose to respect that authority because they respect the person