The difference is caused by the fact that there is an additional Feynman diagram in Bhabha scattering. (The electron process can occur by scattering OR pair annihilation/production, whereas the muon process can only occur by the latter.)
This gives a scattering amplitude (in terms of the Mandelstam variables s,t,u) of
M_mu = (u^2 + t^2)/s^2
and
M_el = (s^2 + u^2)/t^2 + 2u^2/(ts) + M_mu
The cross-section is proportional to the squares of these quantities. In the center-of-mass frame (same as the lab frame for a two-beam collider), s has no angular dependence, and
t ~ 1-cos theta
and
u ~ 1+cos theta
So the relative angular dependence isn't hard to work out (though I don't know it offhand).
(These values are all from Quarks and Leptons by Halzen & Martin)
The rest is algebra. Note that this is a high-energy approximation, which assumes that the particle masses are negligible compared to the beam energy.
(Hope this is what you were looking for!)
