Gerstenhaber versus Schouten In terms of formal definitions, is there any distinction between Schouten and Gerstenhaber algebras?
 A: Having read Schouten once, I can say that Schouten only looked for differential concomitants of tensor fields which are independent of the connection used, and found the Schouten bracket for skew multivector fields. 
(Added in edit: The graded Jacobi identity for it was noted by Nijenhuis.)
This was extended by his student Nijenhuis together with Frölicher to tangent bundle valued forms. along the way they found also the Nijenhuis-Richardson bracket which is purely algebraic (i.e., $C^\infty(M)$-linear in each entry), so it was taken oven to the purely multilinear setting. But everything was for skew symmetric multiplications. 
I also read Gerstenhaber, and I seem to remember that he came independently from the pure algebraic setting, aiming for the associativity of any bilinear product.
There seems to be a nice survey article of Lecomte and DeWilde on this.
Edit: The following is what I remembered. It seems that the whole book is more relevant than the singe article that I remembered. 


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*De Wilde, M.(B-LIEG); Lecomte, P.(B-LIEG)
Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations. Deformation theory of algebras and structures and applications (Il Ciocco, 1986), 897–960, 
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 247, Kluwer Acad. Publ., Dordrecht, 1988.  

