Time- and angle-resolved two-photon photoemission (2PPE) was used to study the electronic structure and ultrafast electron dynamics of the $p$-doped topological insulator ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ and its derivative ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{2}\mathrm{S}$. Our 2PPE experiments directly reveal that the massless Dirac-cone like energy dispersion of topological surface states is realized above the Fermi energy in both materials. The observed bulk conduction bands of ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{2}\mathrm{S}$ are found to be shifted to higher energies as compared to ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$. This shift has, however, surprisingly almost no influence on the electron dynamics in the topological surface state, which proceed on a picosecond time scale.