Here is a comparison of shopping on Amazon Fresh vs the local Seattle Safeway. The items were picked as a normal shopping trip and not for any special reason. Looks like you can save about 7% by shopping at Safeway. Of course, you also have to take in the fact that while both stores offer a delivery service, Safeway offers free shipping for orders over $150, while Amazon Fresh's limit for free delivery is only $30.
View Spreadsheet Here
Using either service, you are likely to save a little bit of money because they allow you to stick to your shopping list much easier. Plus, being able to shop for groceries while watching the latest episode of 24 is a real plus.