看 is to read with eyes.
讀 is to read with mouth (e.g. read out loud-->讀課文(read the texts)), or read with mind (study-->讀化學(study chemistry)).
If you 看一本书, you are probably reading a book as a recreational thing.
If you 读一本书, you are reading a book as you absorb knowledge from it. You might take notes. You might memorize something in it.
The subtle differences suggested by Messquito is for your information. Another example: we usually say 看杂志 (magazine), not 读杂志, because magazines are usually for recreation.
But if it's a book, in casual conversations, people could use either one.