It would seem foolish to say there's no relation between them, since everything is related, but that's kind of what I'd like to say. As Pavel explained, one (the usual Lie bracket on vector fields) has to do with the Lie algebra of symmetries of your space, while the other (Atiyah) has to do with the Lie algebra of symmetries of POINTS of your space, i.e., the Lie algebra of the loop space. For example take $X=BG$ --- then the Atiyah algebra is just the Lie algebra of $G$ itself, while the usual Lie bracket of vector fields vanishes.

A more fancy way to say this is to do a Koszul dual formulation --- replace Lie algebras by the deformation problems they represent, or the formal moduli spaces you build out of them (by taking solutions of the Maurer-Cartan equations). This makes the distinction between the two very clear: the usual Lie algebra of vector fields on $X$ represents the deformation theory of the space $X$, via Kodaira-Spencer theory (take the derived global sections of the tangent sheaf you get a derived Lie algebra representing deformations of $X$). On the other hand the Atiyah bracket represents the space $X$ itself --- i.e., the deformation theory of a point in $X$ (or if you want to vary the point, of the diagonal in $X\times X$). From this perspective it's also clear why one is linear (local in $X$) and one isn't.

In the example of $BG$, we find the formal moduli space associated to the Lie algebra $g$ is the classifying space of the formal group of $G$ (by definition -- there's nothing in degree one to consider solving Maurer-Cartan with, but we can formally exponentiate $g$ to give automorphisms of this trivial solution).

Another fun example is the case of a singular point of a variety - eg a simple node. We're used to the fact that smooth affine varieties have no deformations, so smooth schemes have no local deformations, so to see something out of Kodaira-Spencer theory you need to pass to something global. But if you take something affine but singular like a node then the tangent bundle (taken correctly, ie the tangent complex) has something in degree one even locally, so you CAN see interesting solutions to the Maurer-Cartan equation even locally, which exactly correspond to the existence of local deformations -- in this case, smoothing out the node.