There are similar questions asked, but none address this question so specifically.
It is, quite simply the title:
What are the drawbacks/disadvantages of using a voltage regulator as a constant current source?
Cost is the only one I can personally think of.
If you are asking about the advantages/disadvantages of an integrated circuit voltage regulator, such as an LM317, compared to a discrete transistor design, I would say that the 317 disadvantages are minimum output current and a relatively large minimum differential voltage across the device (input voltage minus output voltage), and the advantage is superior stability over temperature.
I recently used an LM317 as a constant current regulator, and was impressed with its temperature stability. It might not be so wonderful at higher output currents, but in the 20 mA - 50 mA range it was very good.
For a discrete design, consider these: http://confocal-manawatu.pbworks.com/w/page/82281019/Constant-Current%20Circuits
This is sometimes shown another way:
The minimal solution has only one transistor, which can saturate if necessary. This lowers the minimum input-output differential voltage to less than 1 V, at the cost of poorer initial accuracy and temperature drift:
Cheap, low-quality constant-current sink over wide voltage range
One disadvantage I can think of is that maximum compliance voltage is limited by the drop-out voltage of the regulator, and also by its internal reference voltage. For instance, when using the LM317 as a current source:
simulate this circuit – Schematic created using CircuitLab
One the left you see that it regulates current nicely, since the voltage required at the top of RLa, to pass 100mA, is well below the maximum that the LM317 can put there.
On the right, though, we see the regulator is unable to maintain 100mA, because the required voltage would be +14V, which it cannot achieve. The maximum voltage it can apply at the top of RLb is 11.5V.
This difference between the supply of 15V and the maximum output of 11.5V is the sum of the regulator's drop-out voltage and its reference voltage across R1.
A DIY current source like the one below would not suffer this constraint. Even though both setups are fundamentally voltage regulators, their topology is different:
As you can see on the right, this design is quite able to produce a compliance voltage much closer to the positive supply.
What are the drawbacks/disadvantages of using a voltage regulator as a constant current source.
It is possible to create a constant current source from an adjustable linear voltage regulator by taking feedback across a sense resistor and passing that feedback to the ADJ pin of the voltage regulator, like so
simulate this circuit – Schematic created using CircuitLab
For some use-cases this is perfectly fine. However, it has some significant drawbacks for other use-cases.
First, in addition to the voltage drop across the linear regulator itself, there must be a voltage drop across the sense resistor equal to the voltage required for correct operation of the ADJ pin. This is typically 1.25 volts. Now a constant current device is meant to have some voltage drop in order to regulate the current. However, in some cases the voltage drop for this circuit is too high for a particular application.
Second, if the current to be regulated is "high", then the power dissipated by the regulator, and by the sense resistor, will also be high. This will necessitate some form of heat removal. It may be advantageous to have only one pass element need heat removal measures rather than two. Sensing a large current through a resistor that has a 1.25 volt drop on it may require a beefy resistor, and possibly a heatsink. It may (or may not) be advantageous to sense the current with a smaller resistor and amplify the voltage to feed back to the linear regulator. Obviously if the current being regulated is small enough, this will not be an issue.
Third, many voltage regulators have minimum current requirements, and therefore are unsuitable if the current specification for the constant current source is less than the minimum for the voltage regulator.
I have taken your description of "using a voltage regulator to create a constant current source" to mean the simple circuit shown in my schematic. But both of the issues I described for that circuit can be mitigated by making modifications to that circuit. So, unless we compare two different concrete circuits, any rule we lay out is subject to exceptions.
When looking for LED regulators, I have found that current regulators using an external shunt resistor usually have a lower sense voltage (for example 0.2V instead of 0.6V or 1.25V) than voltage regulator.
As all the current goes through the shunt, a lower voltage makes the circuit more efficient as less power is lost in the shunt.
You don't mention if you are looking for linear or switching regulators : Constant current switching regulators can have bad effects when the output is unconnected, real current regulators are protected against open circuit.
