If you've ever wondered why we don't travel through the core of the planet in order to get to the other side, the video above will provide you with an accurate answer. The clip described how gravitational forces would act on a body falling through the planet, and how that body would behave as a result.
The guys at Minute Physics also describe how lacking a molten core would influence the planet's ability to safeguard us against the negative effects of solar and cosmic radiation. One possible consequence would be that polar lights would be visible from anywhere on the planet.
The magnetosphere, the layer of the atmosphere that protects us against radiation, is largely generated by the motions of the planetary core, which is made up of molten hot iron. Without it, life as we know it would be impossible on Earth.