A constant current source supplies exactly the right voltage automatically to maintain the programmed
current.
A constant voltage source doesn't so can only be used with a resistor to turn it's
thevenin equivalent into something a bit more like a current source - sort of.
Since forward voltages vary by perhaps 10% or so at most, you only need say 20%
spare voltage to allow a series resistor regulate the current to within a factor
of 1.5 or so, which frankly is good enough to fool the eye.
To get the best performance from a high power LED (which are expensive)
its worth investing in an accurate constant-current supply (these are usually
boost-converters with current output).