PWM vs current control for LED
For a normal indicator LED, there will be little difference.  However, the steady current will be slightly more efficient at the LED than pulsed current that averages to the same brightness.  This is for two reasons:

The voltage accross the LED is not fixed.  Even though the LED is a diode, the voltage will still increase a bit with increased current.  Therefore, the LED will be less efficient (power out / power in) at high currents, even if the light output stays proportional to current.

Light output is not totally proportional to current.  The light out as a function of current in graph is a straight line at low power but falls off a bit a high power for most LEDs.  Ordinary 20 mA indicator LEDs are usually not run at very high power, so datasheets for such LEDs will often show a linear relationship between current and light intensity.  However, LEDs for illumination are run much close to the maximum the die can handle, so this effect is usually quite apparent.  Look at the datasheet for any power LED and you should see such a graph.



However, looking at just the efficiency of the LED is missing the point.  If you really care about efficiency, then you have to look at the overall system.  Depending on how exactly the LED is being driven electrically, some of the above may not matter.