Graph sparsification and eigenspaces
I am not sure how you measure distance between eigenspaces, but I do not think you can say anything non-trivial here without some assumption that the eigenvalues of $A$ are separated. For example, let $u$ and $v$ be two arbitrary unit vectors, and define $A = I + \varepsilon uu^\intercal$ and $B = I + \varepsilon vv^\intercal$. Then $(1-\varepsilon)B \preceq A \preceq (1+\varepsilon)B$. However, the leading eigenvector of $A$ is $u$ and the leading eigenvector of $B$ is $v$, and, since they were arbitrary, they can be as far apart as you want.