I've not used it (and, as I'm unlikely to join Amazon Prime, I doubt that I ever will) but a quick look at their prices shows that they're much the same as the major 'full range' supermarkets (i.e. ignoring the 'limited range discounters', such as Aldi and Lidl). The products are all big-name brands, except for the Morrisons own-brand products that are offered (which seem to exclude the 'Savers' range).
So, if you're happy to stick with the big-name brands you'll pay around the same as you would with Asda, et al. However if you want the cheapest deals you'll have to shop elsewhere.
e.g. If you want a tin of corned beef, you'll be offered the Princes brand, at £2.00. That's the same prices as in Asda and Morrisons (although cheaper than Sainsbury's [£2.30], Tesco [£2.50], Ocado [£2.50], Waitrose [£2.60]) but still around a third dearer than the price I'd paid for alternative brands in Lidl, B&M or Asda.
Similarly, if its cat treats you want, Dreamies are offered at £1.00. That's the standard price (although they often tend to refer to it as an offer price) in many of the major supermarkets. However you can go into Asda and buy their own brand for 65p, or Poundstretcher and get their brand at 2 for £1.
So, if you're already a Prime member, and you don't mind only being offered just big-name brands, it might be worth giving it a go. However if you're looking for the cheapest prices (irrespective of brands), go elsewhere.