BJT: Equation of Vce for Saturation Region
Why does a difference in junction area causes a change in the
  forward-voltage drop?


Consider the shockley diode equation: -



If a diode has a higher reverse saturation current (\$I_S\$) then, for a given forward volt drop (\$V_D\$) across it, there is more forward current (\$I_D\$). Alternatively, for a given (and constrained) forward current there is a smaller forward volt drop compared to a diode having a lower reverse saturation current.

In heavy saturation the BC junction is starting to forward conduct and, because it has a bigger surface area compared to the BE junction it naturally has a higher reverse saturation current therefore, it has a lower volt drop from base to collector compared to base/emitter.

In other words the collector can never fully get down to the emitter voltage when the transistor is saturated because it is being "held up" (or supported by) the base voltage.