I haven't been to the zoo in years...but it is one of THE best zoos in the world.
The Park? I was last there in 2008.
If you haven't been to either the Zoo or the Park and this is the first time, I would choose the zoo. It really is a terrific zoo and Balboa Park is gorgeous. The zoo is set on and in between two inclines; rather like being in a shallow gorge. So, the setting is quite different.
Check out the museums at Balboa Park and see if there are any you want to do as well, if you have time.
The African animals are at the Park and I can't remember if there are any real close encounters i.e., feeding/petting. But you can do close encounters at Werribee Park Zoo?
The Park is out near Escondido and if you have a car, it should take you about 45 minutes from SD to get there. There are a couple of wineries along the drive close to the Park. I didn't do any of the close encounter safari rides; just the general one and caught the bird show.