rotationangle[{i_, j_}] := Which[
  i >= 0 && j >= 0, VectorAngle[{1, 0}, {i, j}],
  i < 0 && j >= 0, VectorAngle[{0, 1}, {i, j}] + Pi/2,
  i < 0 && j < 0, VectorAngle[{-1, 0}, {i, j}] + Pi,
  i >= 0 && j < 0, VectorAngle[{0, -1}, {i, j}] + 3 Pi/2]

quadrant[{i_, j_}] := Which[
  i >= 0 && j >= 0, 1, i < 0 && j >= 0, 2,
  i < 0 && j < 0, 3, i >= 0 && j < 0, 4]

anglecalc[u_, v_] := Module[{a, theta, r},
  a = VectorAngle[u, v];
  theta = -rotationangle[u];
  r = RotationTransform[theta];
  If[quadrant[r[v]] > 2, 2 Pi - a, a]]

(* case 1 *)
{a, b} = {{{4, 3}, {3, 0}}, {{3, 0}, {10, 1}}};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

r = RotationTransform[90 Degree];

(* case 2 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.22], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 3 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 4 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.22], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 5 *)
{a, b} = {{{4, 3}, {3, 0}}, {{3, 0}, {10, -7}}};
Graphics[{Arrowheads[0.12], Arrow[{a, b}]}, ImageSize -> {120, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 6 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.09], Arrow[{a, b}]}, ImageSize -> {Automatic, 120}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 7 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.12], Arrow[{a, b}]}, ImageSize -> {120, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 8 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.09], Arrow[{a, b}]}, ImageSize -> {Automatic, 120}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 9 *)
{a, b} = {{{4, 3}, {3, 0}}, {{3, 0}, {-7, -5}}};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 10 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.13], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 11 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 12 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.13], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 13 *)
{a, b} = {{{4, 3}, {3, 0}}, {{3, 0}, {-7, 5}}};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 14 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.21], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 15 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.1], Arrow[{a, b}]}, ImageSize -> {150, Automatic}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

(* case 16 *)
{a, b} = r /@ {a, b};
Graphics[{Arrowheads[0.21], Arrow[{a, b}]}, ImageSize -> {Automatic, 150}]
{u, v} = a - b;
N[anglecalc[v, -u] 180/Pi]

myVectorAngle[{a_, b_}, {c_, d_}] :=
 Evaluate[With[{code =
     ReIm[(c + I d)/(a + I b)] // ComplexExpand // Simplify},
   ArcTan @@ code
   ]]