Welcome to CPF, DanMill!
This topic probably belongs in the Electronics forum, where the real whizzes would surely clear the air - but I'll give my $.02...
The problem is that - unlike most component resistance/loads folks are used to - LED's often tend to display the characteristic of requiring LESS voltage as they "heat up", to pass a target current - and the current is what does the damage (creates heat). LED's are current driven devices, and the manufacturers' specs state a very specific current limit, but only a loose range of drive voltages.
So, for example, you drive a single Nichia white LED at it's max recommended current of 30ma, by setting a supply voltage of, say, 3.65v - and let it run on the bench. Come back after 20 minutes, and you may find that same 3.65v is now OVERdriving the LED at 40ma - which produces excessive heat, which further drops the LED's voltage requirement (Vforward, or Vf), which further increases the current, more heat, etc... A chain reaction - thermal runaway, I think they call it around here - that can/will fry the LED. Supplying a source of constant current instead of voltage avoids this problem by simply regulating the voltage downward when this happens.
AFAIK, I think it's really only a problem when pushing currents beyond the manufacturer's specs, or when heatsinking is inadequate. Many of the currently available single LED flashlights use voltage regulating circuits and are often overdriven as well, with little/no ill effects or runaway.
For LED strings in series, it is wiser to use constant current regulation, as LED's of even the same batch can vary a fair amount in Vf requirements - and those variations can compound the current limit/thermal differences of the individual LED's across the string - each having an effect on the other - leading to a greater possibility of thermal runaway somewhere along the string. A constant voltage may be being applied to the entire string, but it's really not controlling the V drop (voltage applied) "seen" by the LED's, individually. I think... (?)
A constant current regulator avoids that issue entirely - what current goes thru the first LED in the string is exactly the same as what goes thru them all.
Driving LED's in parallel, though, is a different story - I think constant voltage is safer. If, for instance, a constant current is supplying 4 LED's - and one blows "open" for some reason (a common failure mode) - the current supply will now deliver it's 4 LED's worth of current to the remaining 3, overdriving them to failure, possibly.
In that scenario, a constant parallel voltage supplied instead will have no effect on the operation of the remaining 3 LED's.
I'm not very experienced in these circuits, I'm just blabbering' a bit of what I've picked up lurking here a while, too. I might be somewhat off the mark - someone please feel free to correct any errors, explain more concisely, and deduct my $.02 if need be!!! /ubbthreads/images/graemlins/twakfl.gif