Using the following definition I don't understand why a reversible process is not spontaneous. For an isolated system, a reversible process happen without any outside intervention.
reversible process: the system changes in such a way that the system and surroundings can be put back in their original states by exactly reversing the process.
Spontaneous processes: those that can proceed without any outside intervention.
A spontaneous process is like a ball rolling downhill. Forces push the ball in the downhill direction. The ball will gain kinetic energy. You would have to put that energy back to reverse it.
A reversible process is like a ball rolling on the level. The ball proceeds without gaining or losing kinetic energy. It could go the other way in the same fashion.
This question can possibly be addressed without invoking entropy, but entropy provides a simple resolution for many-particle systems.
All spontaneous processes require a nonzero driving force to push them to occur at any finite rate. This force can be generalized as a gradient; i.e., a difference in temperature or pressure or surface tension or electric field or chemical potential—any intensive property. (For example, we can't put identical-temperature systems in contact and expect to obtain net heat flow.)
Entropy is generated whenever energy is transferred down a gradient. Entropy is related to the distribution of energies, and the state at the bottom of the gradient gains more distributional possibilities for a given energy than the state at the top of the gradient loses. (To return to the heat transfer example, isothermal cooling of a hot object at temperature $T_\mathrm{H}$ removes entropy $\Delta S=Q/T_\mathrm{H}$ for heat transfer $Q$, but isothermal heating of a cold object at temperature $T_\mathrm{C}$ contributes entropy $\Delta S=Q/T_\mathrm{C}$. Since $T_\mathrm{H}>T_\mathrm{C}$, entropy has been generated in the process of allowing heat to move down the temperature gradient.)
Unfortunately, the Second Law tells us that entropy can't be destroyed; it's too unlikely for the energy distribution of a many-particle system to independently squeeze into a tighter profile.
Reversible processes must therefore generate no entropy, as we would not be able to return the system and surroundings to the original state—there would be an entropy excess.
Thus, a spontaneous process can never be reversible. (One can even show that spontaneity as characterized by free energy minimization implies entropy maximization. For instance, all chemical reactions must either produce higher-entropy products or must heat their surroundings, which increases their entropy.)
Why, then, would we ever consider reversible macroscale processes if they can never occur in reality? Because it's convenient to not worry about that finite amount of entropy being created as a function of the process speed and conditions. (Put another way, it's annoying to work with a one-sided conservation law; energy, in contrast, satisfies two-sided or absolute conservation in essentially all contexts.) And in any case, we could come arbitrarily close to reversibility with careful design and by running a process very slowly, so an analysis assuming reversibility isn't completely ludicrous. (Returning again to the heat transfer example, we assume that a very slight temperature difference drives heat flow spontaneously and with negligible entropy generation in a nearly reversible process.)
