Special Agent (or why is Manganate sometimes used acidified and sometimes in alkaline solution)

Potassium Manganate (VII) is either used acidified (when oxidising alcohols) or in alkaline solution (when oxidising methyl benzene)

The reason why you have to acidify the manganate VII for the oxidation of alcohols is because the reaction needs H+ to combine to make water from MnO4-'s oxygens. As you saw in CH5 when doing redox half equations for Manganate (VII).

The reason why you need alkaline conditions when oxidising methyl benzene - I have no idea. 

I even googled it and found some degree level notes on it and they said the mechanism was to complicated to explain at degree level. So if I was you, I would assume it is wizardry and move on.