Approve means to substantiate, endorse, or sanction officially and render something as good or acceptable. It could also mean favorable or the right thing or action. On the other hand, authorize means to grant permission for, to allocate power to, or to pass. It also means a position of power over others, for good or for bad.
You could even say that authorize is a word of formal action, and it is an extension of authority to another person for some productivity or engagement. Approve means it is something with which you are either okay or on board. Approval means you have been given the authority to achieve or do something. Approval is also something that can be earned. You can run into a position where you have to prove yourself to gain approval; however, it sounds more commanding to have authority over something or someone.
An example of this would be “He has the authority to fire me from my job.” An example of approval would be, “I approve of the two of you entering into this marriage. There are some formal situations in which using the word authority is appropriate and vice versa.