I've moved most of my online purchase of new books to The Book Depository, at the recommendation of peteyoung, saare_snowqueen, and several Facebook friends. It's not without its flaws. Although the prices are usually cheaper, delivery can be slower (not that that usually bothers me, and at least they are up front when they can't deliver at all). I was also very annoyed when I tried to switch my Christmas wishlist to The Book Depository's system. It seems to be impossible to change your wishlist to rank the books other than in order of your adding them, starting with the oldest. This is crazy - the books I am most interested in getting are the ones I've listed most recently, and Amazon's offering of ranking by price or urgency also helps people who want to buy books for you.
Much more seriously, several people to whom I sent the wishlist link - supposedly http://www.bookdepository.com/wishlist/498854/Nicholas-Whyte - were unable to see anything other than the Book Deposityr front page; my mother emailed The Book Depository for assistance and was given the extraordinary answer that "The only way you can access other people's wishlists, is if you have their account details to login" which rather removes the point of having them! So in the end I had to re-circulate the link to my Amazon wishlist, if I wanted to get any Christmas presents.
The other annoying thing that then emerged about The Book Depository's system is that, unlike Amazon, when you buy a book for someone it doesn't disappear from their wishlist. This meant that when books did arrive from The Book Depository, I had to manually delete them from both lists. There was also the obvious risk of two people getting me the same book. So basically, The Book Depository's wishlist system sucks, and clearly hasn't been designed with actual customers in mind.
(The service is also a bit low-end: "We cannot offer any tailored gift options to customers. All of our orders will be accompanied by a packing slip which features the email address of the customer who placed the order, and the cost of the order." So your present won't be wrapped but will have the price tag. Classy.)
I see that The Book Depository also has an affiliate scheme, but I'm not sure if I want to invest the time and effort into adding their links to my website, or any other space I control, until I feel a bit more confident that they are getting their act together. And I'm on the lookout still for competent alternatives.