As for the big gain being that it's right there, I figured everybody's phones are right there too. Apparently not everybody does that. Seems like if you really find this to be a problem, you could just keep your phone in your pocket.
Edit: and the answer is pretty clear, I think. If people actually go for this, it'll be a massive advantage for Amazon. It essentially banishes competition permanently for people who use the button regularly. It'll also make people much less price sensitive. So I completely understand why Amazon would try to push this. I just don't understand why anyone else finds it interesting besides that.
Could apps be better? Sure. Can any app get as good as a button that's already right there when you realize you need a replacement? I don't see how.
It takes more than 10 seconds to go to the store and get detergent. I suppose it'll save the 30 seconds it takes to order detergent on a phone, but the big gain is that it's RIGHT THERE so you can't forget to order it and then find yourself dead in the water and have to go to the store to get detergent instead of wait 2 days for amazon to deliver it.