"Equilibrium" means thermal equilibrium. The solid has one well defined temperature, and a constant Fermi energy. The Fermi energy is an energy value against which energy levels are compared to determine how fully occupied (or not) an energy level is. Generally when the Fermi level is constant throughout a solid electrons diffuse equally in all directions.
"Steady state" means that the properties of the system do not change with time. Non-equilibrium states can be steady states if there is a source of energy to maintain the non-equilibrium condition. Without the source of energy, the system would quickly settle into an equilibrium state.
A p-n junction (electrical diode) with no voltage applied is in an equilibrium state. The same diode with a voltage applied across it is in a non-equilibrium, but steady, state. The applied voltage raises the Fermi energy on one side of the junction relative to the other, and that difference in Fermi energy is maintained by the voltage (energy) source. Electrons diffuse preferentially in one direction. The properties of the system do not change with time, but the state is non-equilibrium.
The analysis of these states is more or less the same in both cases. Solid state physics, statistical mechanics, and transport theory all apply to both cases. The conditions under which the analysis is performed changes, though. For example, in a biased p-n junction one has to set the condition that the Fermi energy changes at the junction.