I would personally choose Brookfield, but both zoos have their positives and negatives. Lincoln Park has an amazing Chimpanzee and Gorilla complex along with a large variety of species, but many exhibits are vastly outdated and small. Brookfield has a large amount of land with a great collection, but as aforementioned, it is farther from the city and cost money to both park and enter into the zoo. Many of Brookfield's enclosures are also much better than that of Lincoln Park. In addition, a dolphin at Brookfield is due to give birth any day now.
Another option is that you could go to the Shedd Aquarium which is also in Chicago. If you are at all interested in aquatic and marine life, choose the Shedd. (If you choose the Shedd, I suggest that you purchase your tickets in advance). If you are more inclined towards terrestrial animals, then Brookfield is your best bet.