The main difference between Earth's mantle and its core is the material making up each section. Temperature and function also differ between the two sections.
The Earth's mantle is made up of semisolid rocks. While different layers of the mantle are made of different kinds and densities of rocks, all are more dense than the crust, and the entire mantle is hot. The temperature of the mantle is hottest nearest the core, and cooler near the crust. The difference in temperature in the mantle accounts for the density of rocks. Near the crust, rocks are cooler and more brittle. They break, as when faults shift and an earthquake occurs. Closer to the core, rocks are molten and flow, as can be seen in lava.
The core consists of extremely hot metal layers instead of rock. Iron and nickel make up the outer section of the core, while the interior is almost entirely iron. The inner core is almost totally solid and shaped like a ball. Despite being the hottest section of the entire planet, it is the most solid because of the tremendous pressure it is under. The core is much denser and heavier than the mantle. The core also creates heat, while the mantle just absorbs it. Heat comes from the radioactive elements in the Earth's core.