As I worked to get my degree in Zoology I was constantly asked "Oh, so you want to be a zookeeper?" Which though I could go down that path if I chose, it was not the reason for my degree.
Getting a degree in Zoology offers up a lot of doors for you. Yes you can become a Zookeeper as a former acquaintance I know currently does with her degree. Though you can also go down the path more related to science or animal welfare. These paths include wildlife science (which is my interest), Veterinary science, Animal behavior, Animal welfare, Rehabilitation centers, and more. Zoology opens up doors to all things animals as the latin roots directly translate into Animal (Zo/Zoa) and Study of (Ology). In essence the two are not mutually exclusive from one another. Though from my understanding, many people can work their way into becoming a zookeeper without having a degree, while you can't necessarily work your way into doing wildlife science, or the other pathways it offers.
I would love to answer more questions for you if anything I said was unclear as it is a common misconception that they are one in the same.
Zookeepers work with wild animals that live in zoos. They maintain the areas where the animals live and ensure they are cleaned regularly. Zookeepers also monitor the animals to make sure that they're healthy.
If you want to study animal habits, what they eat, how they mate and so on, this is a zoologist job. If you want to take care of animals, tend them, feed them ... this is a zookeeper job. The zookeeper probably uses what the zoologists found to best take care of the animals in their parks.
That said, I can be wrong.