Their UI hasn't really changed in years, but their UX has, a lot. Do you think they would have seen the same amount of growth if they'd focused on improving the UI of their site, rather than the UX of their service?
It's not that UI is unimportant. On the contrary, I personally think it's very important, but only in as much as it serves the UX. This for instance means that a company like Apple has to invest a lot more in evolving and improving their UI. It's fundamental to their UX of beautiful, useable devices. In a sense the UI is the UX, so they're tied together very closely. For most companies/products that isn't the case though. Most are more like Amazon.
To Amazon, the UI just needs to be functional, and get the user through to making a purchase as quickly and with as little hassle as possible. That's why they've stuck with essentially the same design and layout for years - customers know it, and it just works. Any changes making it fancier or prettier would be, at least in the short term, detrimental to the UX.