Alan, why not read both then let us all know? ;-) Haha. I tried to read the Erik Braun one, but didn't finish it. Written by an academic, every page is heavy with names and dates and information, and so there's only so many hundreds of such pages one can get through. Still, an important book if you have the time and drive.

Some other recommendations:

Give the Visuddhimagga a try. It's a 5th century CE text from Sri Lanka that was a compilation of all the meditation-related commentaries up to that point. That's around a 1,000 years of wisdom collected by leading lights in the field. As you might guess, it is heavy-going, but think of it as an encyclopaedia - you just look up the topic you want to know more about, rather than read it from cover to cover. It's also one of the central texts of the whole Mahasi method thing in Myanmar. The teachers quote from it, especially in the Sayadaw U Pandita tradition. Once you've immersed yourself in it for a while, you'll realise it's one of the most important books a meditator can have. 

Second, see if you can get a copy of the biography of Mahasi Sayadaw. Some day if you get a chance, you can train at the Mahasi Centre in Yangon where they have a bookshop packed with English translations of loads of books about him and by him. Also a great resource to understand him and his intentions better. The best of them is the biography. It details his life, as well as how he gathered together the threads of scholasticism and practice techniques to come up with the method that bears his name. 

All the best!