Filling between two arrays with ListPointPlot3D

Filling -> {2 -> {1, Directive[{Opacity[0.4], Red}]}}

Update: Since the two surfaces are separated by a plane a much easier approach is to fill both surfaces to a plane (say, the z == 1500 plane) between the two and post-process to remove the polygons whose third coordinates are constant at z:

z = 1500.;     
DeleteCases[Normal[ListPlot3D[{cc, bb}, PlotRange -> {500, 3000},
      Filling -> {1 -> {z, Opacity[.5, Green]}, 2 -> {z, Opacity[.5, Green]}}]], 
  Polygon[{{_, _, z} ..}, ___], All]

Original answer:

David's method fills from the top surface to the z == 1 plane. To fill between the two surfaces we can

1. Create a ListPlot3D object, lp3D, using bb with filling to a plane below the cc surface (say, the plane z==0)
2. Post-process the output of the previous step (a) to replace the coordinates {x_, y_, 0.} with {x, y, w} using w from the entry of cc whose first two coordinates match {x,y}; (b) remove the polygons at the top and bottom (these happen to be in the first group of polygons)
3. Use Show to combine the result from the previous step with ListPlot3D of the two data sets.

lp3D = ListPlot3D[bb, Filling -> 0, 
   FillingStyle -> Opacity[.5, Green], PlotStyle -> None, 
   BoundaryStyle -> None, Mesh -> None];
assocc = Association[{#, #2} -> #3 & @@@ N[cc]];
lp3D = lp3D /. GraphicsComplex[a_, b___] :> 
    GraphicsComplex[a /. {x_, y_, 0.} :> {x, y, assocc[{x, y}]}, b];
lp3D = Replace[lp3D, {a___, {EdgeForm[], ___}, ___, b : {EdgeForm[], ___},  c___} :> 
   {a, b, c}, All];

Show[ListPlot3D[{cc, bb}, PlotRange -> {500, 3000}], lp3D]

Note: If you define disp = Dispatch[{#, #2} -> #3 & @@@ N[cc]]; and use {x, y} /. disp in place of assocc[{x,y}], and {0, Infinity} in place of All above, this method also works in version 9.