Rotating Vectors in space

I think you want a single transformation function for all points, rather than one for each point independently as kglr's answer currently shows.

set1 = {A0, A1, A2, A3};
set2 = {B0, B1, B2, B3};

tF2 = FindGeometricTransform[
        set1, set2
         , TransformationClass -> "Rigid"
         , Method -> "FindFit"
      ][[2]];

Graphics3D[{
   {Red, PointSize[Large], Point[set1]},
   {Opacity[.15, Blue], Sphere[tF2@set2, 2]}
}]

If you are looking for a map R.A[i]~B[i] ,i,0,1,2,3 : First you have to define the rotationmatrix R

 ϕ = {{0, -φ3, φ2}, {φ3,0,-φ1}, {-φ2, φ1, 0}};
 φ = Simplify[Sqrt[Tr[-ϕ.ϕ/2]]]
 R = IdentityMatrix[3] + Sin[φ]/φ ϕ + (1 - Cos[φ])/φ^2 ϕ.ϕ ;

which depends on φ1, φ2, φ3

Now you can calculate the optimal rotation parameters φ1, φ2, φ3

 J = Total@MapThread[(R.#1 - #2 ).(R.#1 - #2 ) &, {{A0, A1, A2, A3},{B0, B1, B2, B3}}]
 sol=NMinimize[J, {φ1, φ2, φ3}]
 (* {2061.5, {φ1 -> -1.01538, φ2 ->1.45343, φ3 -> 1.3975}} *)

The optimal rotationmatrix is

Ropt=R /.sol[[2]]     
(*{{-0.303462, -0.951876, 0.0429177}, 
{0.00554758, 0.043276,0.999048}, 
{-0.952827, 0.303411, -0.007852}} *)

Ropt.Transpose[Ropt] (*test:  should be identitymatrix[3] *)

You can use FindGeometricTransform:

tFs = MapThread[FindGeometricTransform[{#}, {#2}][[2]] &,
  {{A0, A1, A2, A3}, {B0, B1, B2, B3}}]; 

Graphics3D[{Red, PointSize[Large], Point[{A0, A1, A2, A3}], Opacity[.15, Blue],
  MapThread[Sphere[#@#2, 2] &, {tFs, {B0, B1, B2, B3}}]}]