This work is concerned with approximating matrix functions for banded matrices, hierarchically semiseparable matrices, and related structures. We develop a new divide-and-conquer method based on (rational) Krylov subspace methods for performing low-rank updates of matrix functions. Our convergence analysis of the newly proposed method proceeds by establishing relations to best polynomial and rational approximation. When only the trace or the diagonal of the matrix function is of interest, we demonstrate -- in practice and in theory -- that convergence can be faster. For the special case of a banded matrix, we show that the divide-and-conquer method reduces to a much simpler algorithm, which proceeds by computing matrix functions of small submatrices. Numerical experiments confirm the effectiveness of the newly developed algorithms for computing large-scale matrix functions from a wide variety of applications.