What is the difference between a ballistic and cruise missile?

There are several differences. A ballistic missile is rocket powered, takes off straight up, and the warhead falls to its target. A cruise missile is jet powered with wings, like an airplane without a pilot, and it flies following the earth until it reaches its target, whereupon the warhead is detonated.

Ballistic missiles hit their target using a rocket-assisted ballistic parabola and is a form of indirect fire. This means that the warhead is boosted on the upward journey, and uses the momentum from the launch to reach the target. Missiles don't have to be rocket-assisted, but it greatly increases their range. A cruise missile heads directly towards its target using aerofoils like an aircraft to follow a pre-programmed route to the target, the downfall is that the range is significantly smaller.