The common way that Americans use ‘book’ is as shorthand for ‘make a reservation for’. It’s often used in the travel industry. For example I booked a flight to Rio. I booked a hotel for the weekend. Therefore, ‘booking an appointment’ sounds a little redundant, because an appointment is a kind of reservation. ‘Making an appointment’ is clearer, and what I personally would always say. In addition, ‘booking’ is used for businesses where there is an exchange of money. For example, I booked a session with my physical therapist. (I pay to see her) I made an appointment to see my professor. (I don’t)

There is no difference in meaning. "To book" usually refers to a business or accounting situation, and it means to make a record of something... originally, in a book. It is common for the person who is asking for the appointment to use the word "make," and the person who is actually recording the appointment to use the word "book." Me: "I'd like to make an appointment for Wednesday at 3 pm." Business: "Yes, that time is open. I'll book you for Wednesday at 3." "Make" is a very basic word that can be applied to many different situations. You can make bread, you can make coffee, you can make an error, you you can make an appointment. In some cases there are specific words that "go with" specific activities. You can "make bread" or "bake bread." You can "make coffee" or "brew coffee". You can "make an error" or "commit an error." And you can "make an appointment" or "book an appointment." In every one of these examples, "make" is perfectly correct. (But you can't brew an appointment or commit bread!)

Both have same end result. To “book” an appointment commonly means you are doing so ahead of time; “ make” an appointment commonly means you have walked into an establishment for a service & they tell you to get that service you need to “make an appointment”

There's no difference in meaning, but "book" may be slightly more formal. You could also "schedule" an appointment.