Frobenius theorem to solve PDE v.s. other techniques

Question

When is the Frobenius theorem used to prove existence for PDE on manifolds, as opposed to more analytical techniques? I apologize that my question is pretty vague, but it stems from confusion about what techniques are generally used in geometric PDE. Sometimes I see more analytic machinery (from, for instance, Evans' textbook on PDE) applied in coordinates. Others (Lee's books on manifolds/Riemannian manifolds) prove existence to PDE using the Frobenius theorem. Could someone shed some light on when one technique is used versus another, or any similarities/differences between them? (Perhaps I am making a distinction where there really isn't one?)