I think it has to do with the mental model. The Amazon Dash could be the first thing you think about when needing something, instead of your normal grocer, because its right there in your home. On a phone this would be just an application, which is easily forgotten and since its not special purpose hardware it's always more tedious to use then a device that has just this one purpose and those buttons.
Have you ever tried to scan barcodes with a cellphone? For anything other than some sort of "novelty" "let's trying this cool new QR hipster app" type of situation.
