There are a number of different ways in which this could be discussed. However, we can generally say that authority and the legitimate use of violence go together but that these things do not necessarily confer power. We can also say that it is exceedingly difficult to know for sure when an entity has power.
In the great majority of societies today, only the government has what is called the legitimate right to physical force. In other words, only the government is allowed to use violence to secure its goals. For example, it is the police who are legitimately allowed to use force against citizens. This can be connected with authority. We can say that authority is the legal authorization to coerce people.
But this is not necessarily the same thing as power. Power can be seen as the ability to shape a society and this is more often done without violence than with it. It is often done through means other than the exercise of legal authority. For example, a leftist scholar like John Gaventa would say that corporate interests (and those who support capitalism in general) in the United States exercise a huge amount of power. They do so simply by influencing our thoughts to such a degree that violence and coercion are not needed. It is generally unnecessary for the police to use violence to protect the capitalist system. Gaventa would argue that this is because supporters of capitalism have used things like the educational system and the media to socialize us to never even question capitalism.
Thus, authority gives the right to use violence, but power can be seen as the ability to mold a society without even resorting to violence.