The magnetosphere protects the lower levels of the atmosphere from charged particles from the sun and further sources. This isn't just solar flares, but a constant bombardment of energy. Without it the rest of the atmosphere will began to degrade, with specific worry about our ozone layer. Some animals such as birds are shown to relay on the magnetosphere to help navigate. Aside from the changes to ecosystems and climate, the magnetosphere is also responsible for the aurora borealis, so we would miss out on that.
On the plus side, we'd finally have enough data to answer questions about what sort of radiation shielding generational spaceships would need, and exactly how much cancer prevention Earth's atmosphere is providing.