The elliptical and bell curves give optimal solutions to different problems. Obtaining the most efficient wing for a given span leads to the elliptical distribution, whereas the most efficient wing for a given load is given by the bell distribution. For a given central load, the bell or Prandtl wing will have longer span but a lighter structure and reduced tip losses.
For a swept wing the Prandtl type is inherently stable, so there is no immediate need for a tail stabilizer. However practical considerations include CG range, and this is the tailless type's big weakness: unless it is sharply swept, as in a typical supersonic delta, it will need a separate elevator, fore or aft, to apply the large trim changes necessary. For a straight wing an airfoil with reflex camber to give a static center of pressure will also make it stable. But such airfoils are relatively inefficient and wing area must be increased. All this reduces the practical advantage of the Prandtl wing well below its theoretical value.
Roll inertia will also be affected by the increased span, which can impact highly manoeuvrable or very large aircraft.
One way of looking at the Prandtl wing is to see the tip section as a vortex-reducing winglet, which just happens to be blended in. I recall the Airbus A380 chief designer saying that it makes little difference in practice whether the winglet section is horizontal or vertical.
Having said all that, I am a fan of the Prandtl wing (which is really the Dunne-Prandtl wing, as J W Dunne discovered it empirically and flew it as the D.7 twenty years before Prandtl did the maths). Now that NASA have rediscovered it, I would expect it and its design principles to find some modest wider application in the future, with or without a tail. For example in long-endurance electric drones with fixed CG and minimal maneuvring needs.