Given (not-necessarily linear) fibre bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ with structure sheaf denoted $\mathcal{O}_M$, a (not-necessarily-linear) differential operator taking sections of $E$ to sections of $F$ can be defined as a bundle map $D: J^\infty E \to F$ covering the identity on $M$, where $J^\infty E$ is the limit of finite-dimensional jet bundles with some appropriate infinite-dimensional smooth manifold structure. A finite-order differential operator is one which factors through some finite-order jet bundle.

If $E$ and $F$ are indeed vector bundles, we can characterize linear differential operators by requiring that the bundle map $J^\infty E \to F$ is linear.

However, there is an alternative inductive definition of linear differential operators, which seems to work for any space (or topos) $M$ equipped with a sheaf of commutative algebras $\mathcal{O}_M$ over a sheaf of commutative rings $K$ .

For $\mathcal{O}_M$-modules $E$ and $F$, define the sheaf of $0$th order linear differential operators to be $\mathrm{Diff}^0(E,F)=\mathcal{Hom}_{\mathcal{O}_M}(E,F)$. The sheaf $\mathrm{Diff}^{k+1}(E,F)$ of differential operators of order at most $k+1$ is the subsheaf of $\mathcal{Hom}_K(E,F)$ consisting of sections $D$ whose commutators $[D,f]$ with every section $f$ of $\mathcal{O}_M$ lie in $\mathrm{Diff}^{k}(E,F)$. Then the sheaf of differential operators $\mathrm{Diff}(E,F)$ is defined to be the colimit of these modules.

Does the notion of a differential operator of infinite order in this second definition agree with that given via jet bundles?

It seems to me that this is not the case, since every differential operator in $\mathrm{Diff}(E,F)$ will be locally of finite order, whereas a differential operator of infinite order defined as a map out of $J^\infty(E) \to F$ might not be of finite order locally (as far as I can tell; I don't know any infinite-dimensional manifold theory). For instance, on a compact smooth manifold, by construction $\mathrm{Diff}(E,F)$ should just consist of finite-dimensional operators. If indeed my suspicion is right, that leads me to my second question:

Are there any interesting infinite-order differential operators, especially those that are not locally of finite order?

I can think of one natural infinite-order differential operator which is interesting from a more abstract point of view: the comultiplication $J^\infty \to J^\infty J^\infty$ of the jet comonad. I suspect there might be some infinite-order differential operators of more practical interest to physicists as well, though perhaps these might involve dealing with infinite-dimensional manifolds.