This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λ_α (R) and 1 < p < ∞, the operator f (A) - f (B) belongs to S_{p / α}, whenever A and B are self-adjoint operators with A - B ∈ S_p. We also obtain sharp estimates for the Schatten-von Neumann norms {norm of matrix} f (A) - f (B) {norm of matrix}_{Sp / α} in terms of {norm of matrix} A - B {norm of matrix}_Sp and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences ∑_{j = 0}^m (- 1)^{m - j} ((m; j)) f (A + j K). We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f (A) - f (B) to belong to S_q under the assumption that A - B ∈ S_p. We also obtain Schatten-von Neumann estimates for quasicommutators f (A) R - R f (B), and introduce a spectral shift function and find a trace formula for operators of the form f (A - K) - 2 f (A) + f (A + K).