I1=1;I2=2;I3=3;L=1;
A1=Simplify[L*Sqrt[I1(I3-I2)/(I2(I3-I1))]]
A2=L
A3=Simplify[L*Sqrt[I3*(I2-I1)/(I2(I3-I1))]]
B=Simplify[(L/I2)*Sqrt[(I2-I1)*(I3-I2)/(I1*I3)]]
δ=(((Sqrt[I2 (-I1+I3)])-(Sqrt[I1 (-I2+I3)]))/(Sqrt[(-I1+I2) I3]))

Qold[t_]=Module[{L1,L2,L3,Lp,ψ,Q1,Q2,res},
L1=A1*Sech[B*t];L2=A2*Tanh[B*t];L3=A3*Sech[B*t];Lp=Sqrt[L1^2+L2^2];
ψ=t/I2+L ArcTan[Sqrt[(I3(I2-I1))/(I1(I3-I2))]Tanh[Sqrt[(I2-I1)(I3-I2)/(I1*I2^2*I3)]t]];
Q1={{L1*L3/(L*Lp),L2*L3/(L*Lp),-Lp/L},{-L2/Lp,L1/Lp,0},{L1/L,L2/L,L3/L}};
Q2={{Cos[ψ],-Sin[ψ],0},{Sin[ψ],Cos[ψ],0},{0,0,1}};
res=Q2.Q1];

Qnew[t_]=Module[{l1,l2,l3,ψ,c,s,Q1,Q2,res},
l1=A1*Sech[B*t];l2=A2*Tanh[B*t];l3=A3*Sech[B*t];
ψ=t/I2+2ArcTan[(1/Sqrt[3])Tanh[B t/2]];
c=Cos[ψ];s=Sin[ψ];
Q2={{1,0,0},{0,c,-s},{0,s,c}};
Q1={{l1,l2,l3},{-l2,1-((l2^2)/(1+l1)),-((l2 l3)/(1+l1))},{-l3,-((l2 l3)/(1+l1)),1-((l3^2)/(1+l1))}};
res=Q2.Q1];
R={{0,0,1},{0,1,0},{-1,0,0}};

dif[t_]:=N[Qnew[t]-R.Qold[t]];

Plot[Norm[dif[t],"Frobenius"],{t,-10,10},PlotRange->All]