Mechanism of the oxidation of alcohols with KMnO4
$\ce{KMnO4/OH-}$ Oxidation



$\ce{H2CrO4}$ Oxidation


The two main differences are


$\ce{H2CrO4}$ Oxidation occurs in acid, $\ce{KMnO4}$ oxidation occurs in base
The source of the nucleophile in the removal of the metal ester. In $\ce{KMnO4}$ oxidation, the $\ce{Mn}$ ester itself extracts the $\ce{H+}$ from the alcohol carbon, while in $\ce{H2CrO4}$ oxidation the nucleophile is the solvent.