In the SM we can not add fermionic mass terms like $m \overline{e}\_R e\_L$ to the Lagrangian since these terms are not invariant under $SU(2)\times U(1)\_Y$. 

After introducing the Higgs in the unitary gauge $\phi=\left(\begin{array}{c}
 0\\
 v + H(x)\\
 \end{array}\right)$ into the system we break the symmetry and are able to put mass terms into the Lagrangian which look like $$y \overline{\Psi}\_L\phi e\_R$$ with $\Psi\_L$ the doublet left handed fermion field and $e\_R$ the singlet right-handed fermion and the Yukawa coupling $y$. They will provide us mass terms like $m\_e \overline{e}\_L e\_R$ with $m\_e$ dependent on the vev and the Yukawa coupling.

Why do we need these $SU(2)\times U(1)\_Y$-invariant terms if our symmetry is already broken? How is this method better than just putting the mass terms $m \overline{e}\_R e\_L$ in our system by hand neglecting the $SU(2)\times U(1)\_Y$ symmetry?