Define region between two curves

Provide the coordinate bounds in the second argument of ImplicitRegion:

rf = ImplicitRegion[y <= f[x], {{x, 0, 4}, {y, 0, 200}}]

ImplicitRegion[y <= 12 x^2 && 0 <= x <= 4 && 0 <= y <= 200, {x, y}]

rg = ImplicitRegion[y >= g[x], {{x, 0, 4}, {y, 0, 200}}]

ImplicitRegion[y >= 3 x^3 && 0 <= x <= 4 && 0 <= y <= 200, {x, y}]

intersection = RegionIntersection @@ {rf, rg};

Region[#, AspectRatio -> 1] & /@ {rf, rg, intersection}

RegionPlot gives a better picture:

RegionPlot /@ {rf, rg, intersection}

RegionPlot[{rf, rg, intersection}]

Just a direct way to do:

ir = ImplicitRegion[g[x] < y < f[x], {{x, 0, 4}, {y, 0, 200}}];
ra = ImplicitRegion[y > f[x], {{x, 0, 4}, {y, 0, 200}}];
rb = ImplicitRegion[y < g[x], {{x, 0, 4}, {y, 0, 200}}];
rgns = {ir, ra, rb};
leg = First /@ rgns;
RegionPlot[{##}, PlotLegends -> leg] & @@ rgns