Difference between weak and strong formulation of a Cauchy problem solution

Question

While studying the theory of differential equations (only ordinary equations in fact, I'm just getting started), the teacher told us about these two different kinds of formulation. Given a Cauchy problem, say $y(t)'=f(t,y(t))$ with $y(0)=y_0$, the strong formulation tells us that a solution is "strong" if it is $C^{1}$ and solves the Cauchy problem. The weak formulation instead states that a solution is "weak" if it is continuous and satisfies $y(t)=y_0 + \int_{0}^{t} f(\tau,y(\tau)) d \tau$. Then we prove that a solution is weak if and only if it is strong. But I don't understand why they bothered giving two different definitions of the exactly same thing. Why this? Isn't the second just a different writing of the Cauchy problem? I repeat, I'm only getting started, maybe this distinction makes more sense when you go deeper with the theory.