I think by now you know the preferred store, but I just have to join in. My DS (3 on Aug 31!) has decided that Publix is one of his favorite places on Earth (Second to the "Disney Boat," of course!) He is always asking to go to Publix. All the cashiers and baggers know him. I think he feels like a celebrity when we walk in because they all come over to say hi (we usually go in the early morning, before it gets busy.)
There is no comparison! Winn-Dixie is right across the street, but I honestly don't know a single person who shops there.