Booking vs. Reservation

Booking is a synonym of reservation. They are interchangeable.
Booking is the act or process of writing something down in a book or books, eg in accounting.
Reservation is the act of reserving, withholding or keeping back.
Yes they are. Booking is predominantely Br. English and reservation predominantly Am. English.
There is a shade of difference.
A reservation is the arrangement for (a seat, ticket, etc.) to be kept for the use of a particular person - n a restaraunt, you will hear:
- Do you have a reservation for tonight?
In a hotel: 
- I have a reservation for a double room.
A booking is the arrangement to have a seat, room, entertainer, etc. at a particular time in the future: 
- I would like to book a table for tonight.
- I would like to book a room for three nights next month.
- I have booked a seat on the train tomorrow.
- We have a booking for a table tonight.
Both are used in the UK, with these differences.
The pub we booked a table in today asked if we had made a reservation.
- I have a reservation/booking for tonight in the name of Smith.
- We have no listing for that name for tonight.

- But I booked it/made the booking over a week ago.