This seems to work:

FC1alternative[t_] = 
 Integrate[Sum[DiracDelta[t - k], {k, vti}], {t, t - Vscc, t}, 
    Assumptions -> Element[t, Reals]] /. HeavisideTheta -> UnitStep //
   PiecewiseExpand
Plot[FC1[t], {t, -1, 12}, PlotRange -> {-.1, 3.1}]
Plot[FC1alternative[t], {t, -1, 12}, PlotRange -> {-.1, 3.1}, 
 Exclusions -> None]

You forgot to divide by m. Anyway, your "functions" are awfully convoluted in their definition and Plot gets them in the wrong evaluation order. Insert Evaluate and see what happens:

Plot[pwPAMV[tt], {tt, 0, m}, PlotRange -> Full, Filling -> Axis]
Plot[Evaluate[pwPAMV[tt]], {tt, 0, m}, PlotRange -> Full, 
 Filling -> Axis]

Try this simpler example, where it is easier to understand what happens:

f := Sin[x] /. x -> Pi/2 - x;
Plot[f, {x, 0, 2 Pi}]
Plot[Evaluate[f], {x, 0, 2 Pi}]

Here I get different integrals

Vscc = 1.5; f = 0.6;
\[Tau]1 = Vscc; \[Tau]2 = f Vscc; \[Tau]3m = f Vscc; m = 10;
PAI = 0.04819277; \[Lambda] = 0.03150354; PSA = 0.025; ni = 
 IntegerPart[(m - \[Tau]1 - \[Tau]2)/\[Tau]3m] + 2;
vti = Flatten[
   Append[Append[{0, \[Tau]1, \[Tau]1 + \[Tau]2}, 
     Table[\[Tau]1 + \[Tau]2 + (i - 2) \[Tau]3m, {i, 3, ni}]], m]];
F[t_] = 1 - Exp[-\[Lambda] t];
vf = {F[t] + PAI};
Do[vf = Append[vf, 
    F[t - vti[[i]]] + PSA (vf[[i - 1]] /. t -> vti[[i]])], {i, 2, 
   ni + 1}];
pwPA = Piecewise[
   Partition[
    Riffle[vf, 
     Table[vti[[i]] \[LessSlantEqual] t < vti[[i + 1]], {i, 1, 
       ni + 1}]], 2]];

pwtemp1 = 
  Piecewise[{{pwPA /. t -> (t - Vscc), 
     Vscc \[LessSlantEqual] t < 2 Vscc}, {pwPA /. t -> (t - f Vscc), 
     2 Vscc \[LessSlantEqual] t < m}}];

FC1[t_] := Count[vti, x_ /; (t - Vscc) \[LessSlantEqual] x < t];
FC2[t_] := Count[vti, x_ /; (t - f Vscc) \[LessSlantEqual] x < t];
pwtemp2[t_] := 
  Piecewise[{{PSA^FC1[t], 
     Vscc \[LessSlantEqual] t < 2 Vscc}, {PSA^FC2[t], 
     2 Vscc \[LessSlantEqual] t < m}}];

pwPAMV[ut_] := (pwtemp1 /. t -> ut) pwtemp2[ut];

    pl1 = Plot[(pwtemp1 /. t -> tt), {tt, 0, m}, PlotRange -> Full, Filling -> Axis]
    NIntegrate[pwtemp1, {t, 0, m}]
    pl2 = Plot[pwPAMV[tt], {tt, 0, m}, PlotRange -> Full, Filling -> Axis]
    NIntegrate[pwPAMV[tt], {tt, 0, m}]/m

Hello Andre Luiz,

in my opinion the main problem is the absolute complex structure of your code.

Did you consider to use something like this?

Plot[
 {
  Piecewise[
   {
    {0, x < 3},
    {1 + 1.5 x, 3 <= x < 4},
    {SawtoothWave[{0, 1.3}, .9 x], 4 <= x < 9},
    {0, x < 9}
    }
   ]
  },
 {x, 0, 10},
 PlotStyle -> Red,
 PlotRange -> {0, 8},
 Filling -> Axis
 ]

Your integrands seem to be equal

pwPAMV[t] - pwtemp1 // Simplify