I know that it is associated with Prandtl lifting-line theory and that ideal elliptical wing actually meets the condition of my question. Unfortunately I am not good at fluid dynamics and I would like a relatively simple explanation why downwash (and lift) differs along wingspan (of non-elliptical wings). I would like to understand at least the basics and satisfy my curiosity :)
Why should non-elliptical wings not produce constant downwash over span? They do at one angle of attack, given a proper twist distribution so that the circulation distribution over span is elliptical.
However, once the angle of attack is different from that of the elliptical distribution, the change in local circulation varies in proportion to local chord. If the chord distribution is not elliptical, the new circulation distribution due to that angle of attack change is also not elliptical, and so is the change in local downwash angle.
Sometimes it is desirable to not produce an elliptical circulation distribution rsp. have an elliptical planform. If the objective is to minimise drag, it helps to reduce weight and pure lift/drag optimization is too narrow. What should count is the best ratio of lift minus wing weight relative to drag. R. T. Jones wrote a NACA Technical Note back in 1950 in which he looked at this problem analytically. Wing weight goes up when much lift is created near the tips, because this lift will cause a disproportional root bending moment, and the wing spar, which has to carry this bending moment, is a significant part of the wing structure. Therefore, reducing lift at the tips and adding more lift at the root will create a lighter wing for a modest drag increase, resulting in an overall optimum with an almost triangular lift distribution. When compared to an elliptical wing planform, the total wing span of such an optimized wing is bigger for the same overall drag, but this wing will weigh less.
Another example is wing stall. An elliptical wing without washout will have a constant lift coefficient over span, and Reynolds number effects ensure that the wing will stall at the tips first. An unsymmetrical stall will result in an accelerated roll departure: The stalled side of the wing loses lift, drops and sees an increase in angle of attack which worsens the stall condition. This effect gives an elliptical wing poor handling characteristics and is the reason why many GA aircraft have rectangular or trapezoidal planforms of little taper.