In this paper, we investigate the quantum dynamics of wave packets of bound states in the nonlinear Henon-Heiles (HH) system. The time evolution of a wave packet was separated into two processes involving (a) the motion of its first moments and (b) its spreading. On the time scale when wave packet spreading can be neglected, we were able to establish relations between wave packet quantum dynamics expressed in terms of the initial-state population probability P(t) and classical trajectory dynamics. We have demonstrated that the initial states, which are characterized by a periodic time evolution of P(t) over many periods, are related to the five elliptic periodic orbits in the HH system. The decay of P(t) over a few vibrational periods is related to the classical features of quasiperiodic trajectories and can be accounted for in terms of overlap reduction effects. When the center of the initial wave packet is moved further away from the classical periodic orbits, P(I) is characterized by a single initial peak followed by a "noisy" background. There is no qualitative distinction between the "short-time" behavior of a wave packet initiated in distant integrable regions and in classical stochastic regions. For long times, wave packet spreading effects set in. Quantum mechanical estimates for the spreading time τs, which depends on the initial conditions, were provided. For wave packets initiated on periodic trajectories, τs is very long, becoming shorter as one moves away from the fixed points. The quantum mechanical long time (t>τs) behavior of P(t) exhibits small-amplitude noisy time dependence for all initial conditions. This behavior may be described as "quantum mechanical stochasticity" and is related to the universal feature of irregularity of the spectral resolution of all wave packets in a nonlinear system. From this point of view, all wave packets exhibit "quantum stochastic behavior," but there are vast quantitative differences for different initial conditions that are manifested in terms of the values of τs and of the average noisy amplitude. It should be emphasized that this "quantum stochasticity" is not related to classical stochasticity phenomena.