I just harvested (first time ever) a laser diode and I've got no idea what it is and I don't know what the numbers and letters on it are. I've tried to search for them to find a datasheet but I just can't find anything. I'm pretty sure it says LZL, 0440 on it. Here is the picture:
For your first build, it's definitely recommended to get an off-the-shelf driver - though I'll try and explain here what would happen.
You cannot fix the both current AND voltage output of a power supply for a changing load, since these are a function of the load; V=IR, I=V/R
So if you have an theoretical voltage regulated power supply, it will force whatever current is needed to flow in the load to produce a voltage of it's set reference across the load.
The same is true in reverse for a Current Regulated power supply - it will give you whatever voltage is necessary on the output to allow the set current to flow in the load.
Since Laser Diodes have different load characteristics at different points in their operating ranges, and they have a temperature coefficient, it is important to regulate the current in order to protect the device. If you regulate the voltage in a laser diode, the current flow through it will fluctuate with temperature as the power supply, and can exceed safe operating limits for the laser diode device, shortening it's life and in the case of a thermal runaway caused by the negative-temperature coefficient, be permanently damaged.