We tried Google Express when they offered a significant discount coupon on Groupon. It is fine for what it is - you can get quick delivery from the stores that they offer. If for some reason you like the stores that they offer and prefer not to go in person then it might fit the bill.
For me, it offers nothing relative to Amazon. Anything that I want to buy that I don't mind shipped is also available on Amazon at a better price, with likely more alternative products for my consideration, and more reviews.
The Google thing is also kind of quirky right now in NYC. They deliver to Manhattan but not to Brooklyn, which meant that when I used it, I had to have them deliver to the workplace and carry the item home.
It was still a good deal when the promotion is available (I did end up having to "find" something to buy given the deadline for the coupon, which again speaks to the catalog that they offer). Without the discounts there wasn't really any reason so use it over Amazon.