What is the difference between the state-transition matrix and $e^{\int_s^t A(\sigma) \text{ d}\sigma}x^0$?

Question

We have defined the state-transition matrix ($\Phi(t,s)$) as the limit of $M_0(t,s) := I_n, M_{k+1}(t,s) := I_n + \int_s^t A(\sigma)M_k(\sigma,s)\text{ d}\sigma$ with $A$ being a piecewise continuous matrix function. We can show that $\Phi(t,s)x^0$ is the unique solution of the IVP $\dot{x}(t) = A(t)x(t),\;x(s) = x^0$. However, the function $y(t) = e^{\int_s^t A(\sigma)\text{ d}\sigma}x^0$ should also be a solution of the IVP since obviously $y(s) = x^0$ and $\frac{d}{dt} e^{\int_s^t A(\sigma)\text{ d}\sigma}x^0 = A(t)e^{\int_s^t A(\sigma)\text{ d}\sigma}x^0$. Why doesn't this mean that these two functions are the same?