ε = 10^-6;

solw = NDSolve[{w''[t] + (w[t] + w[t]^2) w'[t] == 1/2 (1 + Tanh[100 t]), w[0] == 0,
w'[0] == 0}, w, {t, 0, 2 π}, Method -> "MethodOfLines"];
wsol[t_] := Evaluate[w[t] /. solw]

solG[s2_] :=
NDSolve[{G''[t] + (G[t] + G[t]^2) G'[t] == s2 1/(Sqrt[π] ε) Exp[-(t/ε)^2], G[0] == 0,
   G'[0] == 0}, G, {t, 0, 2 π}, Method -> "MethodOfLines"]
Gsol[t_] := Evaluate[G[t] /. solG[s2]]
approx[t_] :=
 s1 NIntegrate[Gsol[t - τ] 1/2 (1 + Tanh[100 τ]), {τ, 0, t}, Method -> "DoubleExponential"]

tab = Table[Abs[wsol[t] - Gsol[t]], {t, 0, 1, 0.01}];
NMinimize[Max[tab], s1, s2]