No, it is not a special discount day. It is what it is - pick a day of the week when your crap will be delivered to you. Just like your garbage is being picked up once a week, your delivery dude or gal will be making a trip to your neck of the woods once a week. Available as a perk to Prime members… Like it? love it? Hate it?
well, I just checked and that’s not quite how it works. (I mean I really checked… but going to my cart & starting the checkout process). I do now have the option to pick a day — but I can pick any day, Monday-Saturday. So it’s not like the delivery person is " making a trip to your neck of the woods once a week" — because whatever day I choose, my neighbors can choose other days, and of course other neighbors will still order but not opt into the once a week thing.
I do see a benefit to the customer, however, because it does allow scheduling for convenience --and if I do place orders for multiple items, it would be nice if they all arrive at the same time. And it seems to be an option that can be selected or changed for each order --so I still can get 2-day delivery when in a hurry.
My guess is that the main thing is to cut down on the impact of the free 2-day delivery for people who frequently order items. Wouldn’t make much of a difference to me — I place maybe 2 or 3 orders a month, and also use the subscribe & save feature — but in theory they were supposed to be delivering the subscribe items together anyway. (But they don’t… they all trickle in on different days).
I think this is confusing! 
I guess picking a day could be a nice option if for example I am ordering but going out of town for 3 days so I’d rather the package land on my porch the day I’m due home, not the day before I’m due home.
I think this is just the beginning of it. Amazon is likely gathering data on neighborhood preferences by offering to pick a day to test the waters. The Amazon Neighborhood Delivery Day could happen sometime in the future.
I don’t use it. The whole reason I pay for prime is to get things quickly. I order them when I need them. It frustrates me when they don’t meet their delivery guidelines (which, btw, are two days from shipping not ordering, so when they’re late they just tell you it took x number of days to ship. Grrr).
I can only see a benefit if you’re worried about your packages being stolen and want to schedule for a day you know you’re home.