This is a test of MathJax.
Let $E\to X$ and $F\to X$ be two rank-$n$ and rank-$m$ vector bundles, respectively, with sections denoted $\Gamma(E)$ and $\Gamma(F)$. Naively, we would define a differential operator as any operator $D:\Gamma(E)\to\Gamma(F)$ which can be realized in local coordinates as an expression of the form
$$ D = \sum_{|\alpha|\leq r} A_\alpha(x)D^\alpha. $$
It is not clear, however, that such an operator is even well-defined. Let $U\subset X$ be a local neighborhood, the space of local sections, $\Gamma(E|_U)$. We would like $D$ to restrict to $\sum A_\alpha D^\alpha:\Gamma(E|_U)\to\Gamma(F|_U)$.
The restriction of sections to $U$ maps $\Gamma(E)$ to a subspace of $\Gamma(E|_U)$ (and likewise to $F$). More precisely, $\Gamma(E)\to \Gamma(E)/\sim\hookrightarrow \Gamma(E|_U)$, where $\sim$ identifies any two sections of $E$ which are the same over $U$. The first map is surjective, the second is injective.
If a map $D$ is to be locally well-defined, it must be constant on the fibers of $\Gamma(E)\to\Gamma(E)/\sim.$ Here is an example where $D$ is not. Let $X=\mathbb{S}^1$, $E,F=\mathbb{R}\times\mathbb{S}^1$, so that $\Gamma(E)=\Gamma(F)=C^\infty(\mathbb{S}^1)$. Let $\psi$ be a bump function with support contained in the northern hemisphere, $\{e^{i\theta}\ |\ \theta\in (0,\pi)\}$. Let $f$ be any function with support in the northern hemisphere.
Define $D:C^\infty(\mathbb{S}^1)\to C^\infty(\mathbb{S}^1)$ by convolution: say,
$$(Df)(z) = \int_{\mathbb{S}^1} \psi(e^{i\theta})f(ze^{-i\theta})d\theta.$$
Locally, on $U$ the southern hemisphere, $D$ does distinguish between $f$ and the zero function $0$. However, locally, $f$ and $0$ are indistinguishable! So $D$ cannot be localized.
So an arbitrary operator can use global information to determine the image of a section over a point. By contrast, we would only like to consider those operators which use local information to determine the image of a section. With an appropriate definition of "local" and "differential operator," then, by the theorem of Peetre, these are precisely differential operators.
