Power seems to denote the ability to force and/or coerce.
Authority lends itself to the concept of having a "right to be heard".
Power can come by virtue of superior strength, position, or alignments, that puts one in a posture where they can force something or someone else to comply (generally by pain, threat, or fear.) We would be most familiar with this concept in government, but it also extends to the realm of the Holy Spirit, giving us the power to force the seen world into the mold of the unseen.
Authority seems to me to have less to do with a person's position and more to do with an invisible quality in/on/around them. People marveled that Jesus spoke with so much authority. This authority SHOULD be a part of a person's position of power but is often missing. This causes power to move to establish itself abusively, and it creates hollow shells of abuse.
The Father, in a home, has power by virtue of his size and strength. It is in his position.
In his book "The Power Delusion", Tony Campolo points out that as followers of Christ, we must choose BETWEEN power and Love. Pick one. So if a father decides to walk in power and forces compliance, he loses the Love of the children. But when a child violates the Law of Love, the Father uses power to bring restoration until the child learns to self-regulate by the Love in relationships.
However, eventually, the Father is not going to have power over the children. Therefore, as he goes thru life, he seeks authority instead. He seeks the right to be heard by his children when they are grown.
True authority, it seems to me, is built on this triangle:
Take, for example, your car.