Step 1: Bleaching action of chlorine.
Chlorine reacts with water to form nascent oxygen:
\[
Cl_2 + H_2O \;\longrightarrow\; HCl + [O]
\]
The nascent oxygen oxidises the coloured substances to colourless products by chemical change. Since oxidation is permanent, the bleaching action of chlorine is permanent.
Step 2: Bleaching action of SO$_2$.
Sulphur dioxide bleaches substances by reduction in presence of moisture:
\[
SO_2 + 2H_2O \;\longrightarrow\; H_2SO_4 + 2[H]
\]
The nascent hydrogen reduces coloured substances to colourless forms. On exposure to air, the substances regain colour due to re-oxidation. Hence, bleaching action of SO$_2$ is temporary.
Conclusion:
Chlorine bleaches by oxidation (permanent), while SO$_2$ bleaches by reduction (temporary).