When we say that a system is thermodynamically unstable, it means that there is still a state in which the system is expected to have lower energy than it currently has. A thermodynamically unstable system is yet to attain equilibrium hence it can still undergo further chemical processes in order to attain thermodynamic stability.
When we say that a system is kinetically stable, it means that the activation energy or energy barrier for the reaction system is high. Thus reactants are not easily converted into products. The reaction system remains the same for a long while without change.
Finally, when a reaction is thermodynamically unstable (spontaneous) but no change is observed, the reaction is kinetically stable.
