Laser diodes are amazingly sensitive to ESD and surges. The diode's central structure itself often does not fail, but the combination of high current density and high internal optical flux from a turn on surge or overdrive blows one or the other cleaved end face off the diode. These cleaved "facets" serve as the optical cavity resonator mirrors of the laser. If you damage the laser, it usually becomes a dim LED. This is a very common failure mode. Keep in mind the density of photons inside the diode is much, much greater then what emerges.
This has a name in the business, COD, for Catastrophic Optical Damage, which usually occurs in the nanosecond to microsecond time scale.
The fact is, most DC power supplies have a very undefined startup, often with spikes and surges, that cause the simple limiting resistor to be a BAD idea. In fact most current limited bench power supplies under 300$ in price readily spike LDs in constant current mode.
So the diode driver deals with the exponential portion of the LD current curve, prevent spikes, provides a soft start, and limits overcurrent. Most cheap diode drivers depend on the fact that the current source op-amp has a long startup time for its internal current sources, no matter what. This provides an inherent soft start, but only if using a single supply rail.
Yes, you can run a LD like a LED, but unless the DC power supply is PERFECT without startup surges, they quickly die.
If your going to try this, and its a mission critical phase of your rocket, you should consider some sort of active current limiting, even if its a two transistor, three resistor active current source. Then include something like a Lasorb(tm) across the diode to protect it from ESD and overvoltage spikes.
I'm a laser professional. I have a large pile of LDs on my desk at home that have been killed by cheap drivers that disregard startup conditions, and so called "constant current" Ebay bench power supplies that have a 350 uF or so electrolytic cap hanging off their output stages. In the rare cases when we test large LDs on constant current bench supplies, we place a manually switched shunt across them at startup.
Look, every 5$ to 35$ laser pointer even has a current source in them, and for good reason. If the Chinese could depend on the internal resistance of the battery and a cheap resistor to limit the current, they would. One of the worse things you can do is have moderate to large inductances anywhere near the laser VCC.
You'll see COD sooner or later, even with the most carefully chosen low inductance film resistor. Most LDs have a current risetime measured in nanoseconds, and it takes them a while to stabilize internally. Until close to stable, they are very vulnerable to COD.
I'm hesitant to send you to laserpointerforums.com, the signal to noise ratio there is very bad, but you'll find a variety of low cost, light weight, laser diode drivers advertised there. Some of which are of quite dubious quality.
Three other things:
No cheap, lightweight circuit is failsafe. If the LD is hot enough to burn fishing line, you need to consider eye safety and eye protection for those recovering the rocket.
It only takes about 15-20 mW of laser light to get to the low end of the recorded eye damage threshold in a perfect case. You'll have a lot more then that to get a clean burn of fishing line. For something like this, an experienced laser user, pro or hobbyist, would design a beam containment box or some form of beam stop.
Back reflections of the laser beam into the laser diode are the second leading cause of diode failures. Make sure your design features adsorptive, not reflective, material around the lens focal point.
Laser safety glasses are far, far, cheaper then blindness.
Steve