Why do the elliptical and the rectangular wing show different aerodynamic efficiency?
It turns out, elliptical lift distribution is not the most efficient planform. It is, however, the most efficient planform in terms of L/D for any given span length, and the reason has to do with its constant span induced downwash (see For the elliptical wing, what is elliptical, and why is drag regularly distributed?). 

If you relax the span length requirement and instead optimize with regards to root bending moments, the Bell-Shaped Lift Distribution is actually superior to elliptical (see derivation and NASA paper). For the same root bending moment and the same lift, the bell-shaped distribution has less induced drag.