By definition, a Homogeneous first order operator on the algebra of differentiable functions is an operator like $$ Df = \sum \phi_i \frac{\partial f}{\partial x_i} $$ where $\phi_i$ are differentiable functions on $\mathbb{R^n}$ or open subset of $\mathbb{R^n}$ , satisfying the Leibniz rule.
And we define Homogeneous first order differential operator as a sheaf homomorphism from $\mathcal{A} \rightarrow \mathcal{A} $ , ($\mathcal{A(M)}$ is a sheaf on a differentiable manifold $M$).
Now, I suppose any homogeneous first order operator can be defined as a sheaf homomorphism from $\mathcal{A}$ into itself satisfying Leibniz rule. So what is the difference between above defined two operators? What is the differential structure defined on the second operator that makes it different from the first one?