Peano's existence theorem is as follows:
If: $$(t_0, y_0)\in \mathbb{R^2},\quad R = {\{(t, y) : |t-t_0|\leq a, |y-y_0|\leq b}\}$$ $$F: \mathbb{R} \rightarrow \mathbb{R} \quad cts.$$ $$|F(t, y)| \leq M, \quad M\in \mathbb{R}$$ $$a_* = min{\{a, \dfrac{b}{M}}\}$$ Then, the system $y'(t) = F(t, y(t)), \quad y(t_0) = y_0$ has a unique solution $(t, y(t)): [t_0 - a_*, t_0 + a_*] \rightarrow [y_0 - a_*, y_0 + a_*] $.
Now, it seems that the only difference between this and the Picard-Lindelöf theorem is that the Picard-Lindelöf theorem requires such an $F$ to also be Lipschitz continuous, but it gives that such a solution $y$ must be unique. Is this the main difference or am I missing some big point?