Lincoln Park Zoo and Brookfield Zoo were rivals for many, many years, and directors Marlin Perkins and either Edward or Robert Bean had a strong and very personal distaste of each other. I don't know when the attitude changed but I know by the eighties, LPZ director Lester E. Fisher was a good friend of George B. Rabb of Brookfield Zoo and they often had breakfast or lunch together. They trade animals amongst each other
Lincoln Park is free, small and more conveniently located in the city. It takes a much shorter time to stroll through and see most key species, and it's right on Lake Shore Drive by locations like the Museum Campus. And again, it's free - people who aren't even very interested in animals will stroll in to enjoy the grounds much like a public park. It's also much easier for busy city parents to bring their kids there. They also have more big cats (in worse enclosures...) and much more intimate primate habitats.
Brookfield is not truly in Chicago as far as locals are concerned and I know both locations are in opposite directions from my home. It charges an entry fee and is much roomier and a significantly bigger walk. You can't see everything in a day either, which might detract from some casual zoogoers. There is not as much to do in the area surrounding the park even though you can spend a whole day in there. They have had a number of unique species though such as Okapi, Grizzly Bears, tapirs, orangutans, and sea lions, none of which LPZ currently has.
As a zoogoer, I prefer Brookfield in most cases and have a membership there, but I think for the downtown parent who wants to shut their kids up for a day, LPZ is a better value. It's free, feels closer to home, and you can breeze through it if you want to. I believe Lincoln Park has a third higher attendence as well, probably owing to it's location and price tag.