In the Heisenberg picture we have an equation for the evolution of operators.
dA_H/dt = (∂A_S/∂t)_H +  (iħ)^-1[A_H,H_H].
The corresponding equation in the Schroedinger picture describes the evolution of the mean value of an observable.
(d/dt)<Ψ_s(t)|A_s|Ψ_s(t)> = <Ψ_S(t)|∂A_S/∂t|Ψ_S(t)> + (iħ)^-1<Ψ_S(t)|[A_S,H_s]|Ψ_S(t)>.
(d/dt)<A_s> = <∂A_S/∂t> + (iħ)^-1<[A_S,H_s]>.
If A does not explicitly depend on time and [A,H] = 0, then the mean value <A> is constant.

Let  H = P²/(2m) + U(R).
Then  (d/dt)<R> = (iħ)^-1<[R,H_s]> = (iħ)^-1<[R,P²/(2m)]> = <P>/m.
Then  (d/dt)<P> = (iħ)^-1<[P,H_s]> = (iħ)^-1<[P,U(R)]> = -<∇U(R)>.
These two equations express Ehrenfest’s theorem.

<R(t)> denotes the position of the center of the wave packet at time t.
Is the trajectory followed by the center of the wave packet the same as predicted by the laws of classical mechanics?
md²<R>/dt² = d<P>/dt = -<∇U(R)>.
The classical force at the position <R(t)> of the center of the wave packet is  F_cl =  -∇U(r)|_r=R.
In general  ∇U(r)|_r=R ≠ <∇U(R)>, which is the average force over the whole wave packet.
Therefore the trajectory of the center of the wave packet is in general not the one predicted by classical mechanics.  However if the dimensions of the wave packet are much smaller than the dimensions over which U(r) or changes appreciably, we reach the classical limit.