These two theorems are quite similar, they both show that an IVP has a unique point. Is there any difference between them.
To show an IVP has a solution, I usually show that the map is a contraction, thus IVP has a unique solution by Banach contraction principle. But if we apply global Picard theorem, there is no need to proving T is a contraction (i.e. as long as T is Lipschitz, it's fine.