Rocket and missile seem to be used interchangeably. “We fired a missile at an enemy base.” seems to be about the same as “We fired a rocket at an enemy base”. Given this ambiguity, it makes me wonder how exactly what the difference between the two words is.
Rocket is defined as:
a cylindrical projectile that can be propelled to a great height or distance by the combustion of its contents, used typically as a firework or signal.
Missile is defined as:
an object that is forcibly propelled at a target, either by hand or from a mechanical weapon.
So it seems the key difference between a rocket and a missile is that a rocket is propelled by some kind of fuel while a missile is a more generic term meaning anything flying through the sky.