While I have been aware of the possible conflict between power and authority for some time. It isn't something I have yet examined. Until now!
Wikipedia, Authority, "This article is about authority as a political concept."
In the fields of sociology and political science, authority is the legitimate power of a person or group over other people. In a civil state, authority is practiced in ways such [as] a judicial branch or an executive branch of government.
In the exercise of governance, the terms authority and power are inaccurate synonyms. The term authority identifies the political legitimacy, which grants and justifies the ruler's right to exercise the power of government; and the term power identifies the ability to accomplish an authorized goal, either by compliance or by obedience; hence, authority is the power to make decisions and the legitimacy to make such legal decisions and order their execution.
Unfortunately, the Wikipedia article Power (social and political) is written from the social sciences perspective and does not clearly distinguish between power and authority in politics.
The term authority is often used for power that is perceived as legitimate or socially approved by the social structure.
For the purpose of tagging questions, authority should be used for questions about legitimate power exercised by government. While power should be used when the question is about power exercised by individuals in government where the exercise thereof is by influence, persuasion, force or other means not, or that may not be, set forth in statutes or rules (not legitimate).
Thus How much power does Putin have in Russia? is a power question. As is What powers does the speaker of the House of Representatives have?.
However, What governmental power does the Monarch hold in Great Britain in practice? is an authorty question.
The implications of power are so broad that a precise usage may not be possible, therefore the proposed excerpt for power:
Use when the [authority] tag is inappropriate.
While it is possible to provide a usage excerpt for power, it still may not be useful as a tag.
90 questions contain both "power" and "authority"
1048 questioms contain "power" but not "authority"
343 questions contain "authority" but not "power"
Questions containing search terms in the title may be a better indicator.
187 questions contain "power" in the title
51 questions contain "authority" in the title ("palestinian authority" questions were excluded)
From What are tags, and how should I use them?,
A tag is a word or phrase that describes the topic of the question. Tags are a means of connecting experts with questions they will be able to answer by sorting questions into specific, well-defined categories.
Tags can also be used to help you identify questions that are interesting or relevant to you.
In this case, neither power nor authority are useful for the intended purposes: "connecting experts with questions" and "identif[ing] questions that are interesting". Indeed, for both tags to be useful would require extensive effort and likely expansion to at least six tags.