Australians use both words. It depends on the situation. The word "reservation" is usually used for hotel or restaurant bookings. The word "book"is mainly used to purchase train/plane/bus tickets or ring up for a taxi. Sometimes "reserve" means to save a seat.
Correction: Which word is used most frequently, "to book" or "reservation"?
Examples: Can I make a booking /reservation for two at 8pm for dinner at your restaurant?
I'd like to book a first class seat on the train/plane to Paris.
Would you like me to reserve [ save] a spot for you at the conference?
I'd like to book you as our next guest speaker.
I'd like to book a taxi for 6pm to take me to the airport.