${\mathrm{CO}}_3^{2-}$and ${\mathrm{SO}}_3^{2-}$  cannot be distinguished by :

Lime water cannot discriminate between ${\mathrm{CO}}_3^{2-}$ and ${\mathrm{SO}}_3^{2-}$

The following reaction explains the discrimination,

$\mathrm{CaO}+{\mathrm{CO}}_3^{2-}+{\mathrm{CO}}_3^{2-}\to {\mathrm{CaCO}}_3$ (white turbidity) 

$\mathrm{CaO}+{\mathrm{SO}}_3^{2-}\to {\mathrm{CaSO}}_3$ (white turbidity) 

Since ${\mathrm{SO}}_3^{2-}$ is oxidised by ${\mathrm{H}}_2{\mathrm{O}}_2$ and ${\mathrm{Cl}}_2$, while ${\mathrm{CO}}_3^{2-}$ is not, Hence, lime water cannot discriminate between ${\mathrm{CO}}_3^{2-}$ and ${\mathrm{SO}}_3^{2-}$