What is the principal challenge to making an incandescent-like dimmable LED lightbulb?
The light output of an incandescent bulb depends upon its temperature (black body radiation). This is not the case with LEDs that emit photons of a particular energy - more photons more light. 

Depending on the method used to control the light output you will get different results.

PWM generally gives better control and works well with LEDs. Some controllers may simply be limiting current by controlling the series resistance or the voltage supplied. 

PWM with incandescent bulbs works well up to the point so little power is being fed into the bulb that it fails to achieve sufficient temperature to radiate light - most of the energy is now infra red.