Two different words are translated as either "authority" and "power" in Christ's words. Different translations may have different standards, but there seems to be little rhyme or reasons about which of two Greek words the various English translations are translating as "authority" and "power." It is one of the many cases, where the English reader reads different words referring to the same Greek word and the same word, referring to two Greek words.
The Greek Words for These Ideas
The most common of these words is ἐξουσίαν (exousia) which means "control", "the power of choice", "permission", "the power of authority", "the right of privilege", "abundance of means," and "abuse of power." However, it is also translated as authority as well. It is used to describe both Christ’s power to forgive sins (let go of mistakes) and his power over all flesh (all physical reality).
The other is δυνάμεις (dynamis), which means "power", "might", "outward power", "influence", "authority", "elementary force", "faculty", "capacity", "craft", "art", "capable of existing", "capable of acting", "worth," and "value."
It should be noted that the exousia is used all the Gospels, but dynamis doesn't appear in Christ's words in John. There are, however, two words related to dynamis that are used. By far, the most common is the verb, dynamai, which means "to have power by virtue of your own capabilities", "to be able," and "to be strong enough." There are a number of noun forms, all of which refer to power. It is the word usually translated as "can," but it has much more of a sense of power than "can" in English. Another noun from the same root is from dynatos, which means "strong", "mighty", "possible," and "practicable."
Christ uses dynamis where referring to a display of power or power from innate capabilities. He tends to use exousia to referring to power that given to someone in authority, the power of kings and the power the Father has given him.
Perhaps in today's English, dymanis could be translated "capabilities" and "displayed capabilities." Perhaps more casually, this might be renders as "what someone can do" and "showing what someone can do." For example, in Luk 5:17, the Gospel writer (not Christ) talked about Christ's dynamis, that is, the power to heal. This is a display of power. Dynamis is used when talking about a display of power or ability. Healing is such a display. This is the source of our word “dynamic”. Think of it as more dramatic, to make a play on words.
Exousia could be translated "authority" or more generally as "control." Today, we tend to talk more in terms of who has control over something than who has power over it. For example, in Luk 5:24, a few verses after the example above, Christ refers to his authority over sin using exousia. Forgiving sins is not a display. It cannot be seen. It is a matter of internal authority or ability, hence, exousia, which people have to trust, at least initially.
For example, one can claim exousia in translating Christ’s Greek but people like to see some dynamis to believe it.