All LEDs really have different characteristics, even LEDs from the same reel, even from the same piece of crystal. They can be binned, which does better, but you will still have variations.
If you carefully measure the point at which each individual LED lights up to the same level (place it next to a "standard" LED that is lit by a power supply and a series resistor set to -- do not use a battery, as the voltage may change more than you want over the course of your testing.)
The LED which illuminates last will have a higher forward voltage when it reaches the illumination of your standard.
Think of the water analogy. The LEDs that light up are like sprinklers with their orifices at a lower height, so as you turn the water spigot slowly on, the water finds the easiest path, and comes out of the lower sprinklers. The one up on the hill gets nothing until you increase turn the spigot on even more, and the pressure "reaches" the lone sprinkler up on the hill, where it finally starts emitting water.
I expected this behavior with the LEDs in parallel, and was surprised at the behavior with the LEDs in series, but it happens in both cases. My explanation is that the energy being expended at the LED has to overcome the band-gap in order to emit light, and that energy gap slightly varies from LED to LED.
Full disclosure, I'm still not a full EE, so hopefully this explanation will be corroborated by other answers.
Finally, the Joule thief circuit handles this "problem" somewhat by providing pulses, and where the pressure (voltage) is high enough, the "dimmer" LED still lights up, but more dimly. I've even had LEDs where I could tell that the bonding wire was loose, and the Joule thief made it light up fairly evenly. This should work with PWM as well because PWM uses the maximum voltage and turns it either all-on or all-off, and the high pressure (voltage) of the all-on part should (usually) turn all of the LEDs on.
Commenter Fredled said:
Your explanation are exact but it would be surprising to see such a
big difference in behavior from LEDs "exactly the same" like staying
completely off while others are all completely on. You could expect,
at worse, one led a bit dimmer than the others. – Fredled
At the boundary where it happens, the relatively small differences get greatly magnified. To me, this seems very similar to what happens with a constant current going into a long-tailed pair, making a very small difference in current greatly magnified, making a comparator. If using a two discrete transistors long-tailed pair, any poor matching, or temperature difference, should be very evident. Also accentuating the effect is our own eyesight, when our pupils dilate in the dim light.
Also, I have found this to be common -- the rule, rather than the exception to the rule. Take any dollar store LED light bulb, dim the lights, and light up the circuit board until they just start to illuminate, and you will see variations in brightness, and frequently significantly so.