Like Grigory M says, look up the Hochschild-Kostant-Rosenberg theorem.
This states that for a smooth algebra $A$, the Hochschild chain complex is quasi-isomorphic to the chain complex $(\Omega^\bullet_A, d=0)$ of differential forms with zero differential.
There is another version of HKR which states that, again for smooth $A$, the Hochschild cochain complex is quasi-isomorphic to the cochain complex $(\Lambda^\bullet T_A, d=0)$ of polyvector fields with zero differential. So here you see (poly)vector fields (i.e. derivations), so maybe that gives you one connection to differential operators.
Now in this second version of HKR, these are actually more than just chain complexes but dg Lie algebras. The formality theorem of Kontsevich -- it's in his deformation quantization paper -- says that, as dg Lie algebras (or $L_\infty$ algebras) the two sides are still quasi-isomorphic.
Moreover, if you look at the deformation quantization paper, you'll notice that Kontsevich's definition of Hochschild cohomology is not the "standard" definition (that is, Hochschild's original definition) involving $\operatorname{Hom}(A^{\otimes n}, A)$. Instead, he takes the subcomplex of the Hochschild cochain complex $\operatorname{Hom}(A^{\otimes n}, A)$ consisting of those maps which are polydifferential operators. [However, see e.g. the paper "The Continuous Hochschild Cochain Complex of a Scheme" by Yekutieli for comparison of different definitions of Hochschild cohomology (both for algebras and more generally for schemes).] Then he shows that this thing is quasi-isomorphic to $(\Lambda^\bullet T_A, d=0)$ (as a chain complex and as a dg Lie algebra). So that gives you another connection to differential operators...