Just to elaborate on Scott's answer: the affine Grassmannian is an algebra in an approrpiate sense, which endows sheaves on it with a rich structure which does not exist for the affine flag variety -- in fact the affine flag variety is naturally viewed as a module for this algebra. The easiest (and most fundamental) manifestation of this algebra structure comes from the identification of the affine Grassmannian with based loops in the compact form of G. As such it is a group (not algebraic) -- but even more importantly, it's a double loop space (double based loops in BG -- which is the topologist's version of the algebraic identification of the Grassmannian as moduli of G-bundles trivialized away from a point).

Double loop spaces are "slightly commutative homotopy groups" (aka $E_2$ spaces, or spaces with a braided multiplication), while affine flags don't admit this structure. There's a precise analogy

$E_2$ algebra: topological field theory :: vertex algebra: conformal field theory,

and indeed Beilinson and Drinfeld gave an algebraic form of this $E_2$ structure known as "factorization space" ( a nonlinear version of a vertex algebra). In physics $E_2$ is the analog of saying the structure underlying operator product expansions for local operators in a two-dimensional quantum field theory. This structure is the most basic ingredient in the geometric Langlands program. It's in particular the reason for the commutativity of Hecke operators in the story, which is the first step towards even imagining there should be a geometric Langlands story.

As for the module structure of affine flags over the affine Grassmannian you can read about it in Gaitsgory's paper.