Saturn's solid core is bigger than the Earth. Saturn's core is small though, compared to Saturn's overall size. The core takes up almost one quarter of Saturn's diameter. Earth's inner and outer cores take up over half of its diameter. Surrounding Saturn's core is a liquid mantle that takes up about three quarters of its diameter. Earth's mantle is rock. It takes up a little under half Earth's diameter.
Earth has a thin rocky crust above its mantle. Saturn doesn't have a crust. Above Saturn's mantle is an atmosphere that is very different from Earth's. The atmosphere on Earth is a thin layer. Saturn's atmosphere is much deeper.