Polymer translocation through a nanometer-scale pore assisted by chaperones binding to the polymer is a process encountered in vivo for proteins. Studying the relevant models by computer simulations is computationally demanding. Accordingly, previous studies are either for stiff polymers in three dimensions or flexible polymers in two dimensions. Here, we study chaperone-assisted translocation of flexible polymers in three dimensions using Langevin dynamics. We show that differences in binding mechanisms, more specifically, whether a chaperone can bind to a single site or multiple sites on the polymer, lead to substantial differences in translocation dynamics in three dimensions. We show that the single-binding mode leads to dynamics that is very much like that in the constant-force driven translocation and accordingly mainly determined by tension propagation on the cis side. We obtain β≈1.26 for the exponent for the scaling of the translocation time with polymer length. This fairly low value can be explained by the additional friction due to binding particles. The multiple-site binding leads to translocation the dynamics of which is mainly determined by the trans side. For this process we obtain β≈1.36. This value can be explained by our derivation of β=4/3 for constant-bias translocation, where translocated polymer segments form a globule on the trans side. Our results pave the way for understanding and utilizing chaperone-assisted translocation where variations in microscopic details lead to rich variations in the emerging dynamics.