The difference is this:  In a reversible process, when you restore the system to its original state, the surroundings are also restored to their original state (note that the entire cyclic process, including the movement away from the initial state, and the restoration to the initial state, must be reversible).  Hence all the state functions and state variables of both the system and surroundings are returned to their original values. 

In an irreversible process, when you restore the system to its original state, the surroundings are not restored to their original state.  Hence, while the state functions and state variables of the system are returned to their original values, not all the state functions and state variables of the surroundings return to their original values.

**I.e., what distinguishes reversible and irreversible processes is not the final state of the system, but the final state of the surroundings.**

Focusing on the 2nd law:

In a reversible process, when the system is returned to its original state, $\Delta S_{system}=0$ (because S is a state function) and $\Delta S_{surroundings}=0$ (because the process is reversible).  Hence $\Delta S_{total}=0$.

In an irreversible process, when the system is returned to its original state, $\Delta S_{system}=0$ (because S is a state function) but $\Delta S_{surroundings}>0$. We know that $\Delta S_{surroundings}>0$ because the 2nd law requires that, for an irreversible process, $\Delta S_{total} > 0$.