Stability is a term which makes a distinctions between different types of equilibrium, for example, take a sphere resting on a flat surface - move it and it will move to another place on the surface and its potential energy will remain the same. On the other hand, take a sphere placed on the peek of let's say some parabolic shape, assume it's in equilibrium - then apply a force to it - it will fall down to the bottom, it's kinetic energy will grow and it's potential energy will become smaller. Last, take a sphere placed in let's say some kind of crater at it's lowest point, remove it from that point (pushing it towards the upper part of the crater), and let it go - after some motion, it will end up at the bottom of the crater again. These three types of equilibrium are called indifferent, unstable and stable, respectively.
Another example would be to look at a cone placed on its lateral surface, on its top and on its base, respectively.
So, the term stability describes different aspects of equilibrium - not that in all the cases the equilibrium equations are satisfied, so the term "equilibrium of a body" needn't give much information in some certain cases.