I have used both Cozi and Groceries IQ. Cozi is by far a more comprehensive app. If you are strictly wanting a grocery list then Groceries IQ is easier (barcode scanning), but with the calendar and the widget capabilities, Cozi just can't be beat. I will be at home while my husband is in walmart and i will add to the grocery list. it updates instantaneously. you know... those last minute items you forget to add to your list. this has helped us reduce our grocery shopping trips for items we forgot! helped us save on grocery bill!
I've used GroceryIQ for some time now and like the synching between my wife's phone and mine, along with the barcode scanning to add items, shopping cart totals, and the fact that it links with Cellfire to let you add coupons. The app is also "powered by" the Coupons.com website.
I've also used it at locations like Lowe's, Wal*Mart, BestBuy etc. so it's not just good for Groceries.
To me, the calendar is Cozi wasn't a big seller because we already share our Google calendars & I didn't need a grocery widget. But other than that, they both look to do the same thing. Cozi just has a paid ad-free version while GroceryIQ is ad-free with free version. But both seem to be strong, well thought out apps.
Another decent one is OurGroceries. Paid app comes ad-free as well and seems to have pretty much what the other two mentioned have.