  Departing from the weak solution, we prove the uniqueness, smoothing
estimates and the global dynamics for the non cutoff spatially homogeneous
Boltzmann equation with moderate soft potentials. Our results show that the
behavior of the solution(including the production of regularity and the
longtime behavior) can be {\it characterized quantitatively} by the initial
data at the large velocities, i.e.(i). initially polynomial decay at the large
velocities in $L^1$ space will induce the finite smoothing estimates in
weighted Sobolev spaces and the polynomial convergence rate (including the
lower and upper bounds) to the equilibrium; (ii). initially the exponential
decay at the large velocities in $L^1$ space will induce $C^\infty$
regularization effect and the stretched exponential convergence rate. The new
ingredients of the proof lie in the development of the localized techniques in
phase and frequency spaces and the propagation of the exponential momentum.
