Why is bleaching with chlorine permanent but with sulphur dioxide temporary?
The bleaching action of Chlorine is permanent because it involves the process of oxidation. $\ce{Cl2}$ reacts with water to produce nascent oxygen which combines with the material in question (by oxidising it, as pointed out by @Mathew Mahindaratne
) and makes it colourless.
The bleaching action of $\ce{SO2}$ is temporary because it involves the process of reduction. $\ce{SO2}$ removes oxygen from the coloured substance to make it colourless. Atmospheric oxygen slowly takes place of the removed oxygen, because of which the material regains it's colour.