5 Fe²⁺ + MnO₄^- + 8 H⁺ → Mn²⁺ + 5 Fe³⁺ + 4 H₂O

5 C₂O₄^2- + 2 MnO₄^- + 16 H⁺ → 2 Mn²⁺ + 10 CO₂ + 8 H₂O

The reduction of permanganate is common to both reactions so that can't be an important contributing factor to the difference in rate. The oxidations have been indicated in blue. In the first case there are no bonds broken. Only an electron is lost. But when oxalate ion is oxidized it must split (in oxalate there is a bond between the two carbon atoms giving roughly this structure: O₂C-CO₂). The energy required to break this bond is probably a contributing factor in the slower rate since collisions must have the energy needed to cause this change.