I did alot of research on this, and the best understanding I found was like so:
Ultimately, permanent means you killed the hair follicle. Anything which does not do this (shaving, waxing, etc.) is short term/cosmetic.
Electrolysis does this by inserting a wire, and electrocuting the follicle. Electrolysis may not work (per follicle) if the follicle is dormant, or if the wire is not inserted properly (doesn't reach the follicle) or if the voltage is not sufficient. Surface scarring may result from over-voltage or improper insertion. But most people have a vague idea how electrolysis works, yes?
Laser still has to kill the follicle to be effective. It does this by exciting (adding energy to) the molecules in the hair. Assuming the follicle is non-dormant, the hair is attached to the follicle and energy added to the hair will be transferred to the follicle. If enough energy is applied, the follicle dies. Laser applies this energy through emission of stimulated electrons in a wide beam through the skin. These electrons will interact with pigment molecules. This is why laser hair removal works best on people with fair skin and dark hair. If you have dark skin, the laser will warm your skin, possible never reaching the hair. If you have light hair, the laser won't excite the hair enough. Beyond that, the same ideas behind the problems with electrolysis apply. If not enough energy is applied to the hair, you may damage the hair currently growing but not kill the follicle. If too much energy is applied, you can damage the skin (and possibly still not kill the follicle, depending on pigment contrasts).
Laser and electrolysis have the same limitation of only being able to kill active follicles, so a proper plan of repeated visits is appropriate. This was excellently covered at
tsroadmap but I can't find the exact text, at the moment.
Finally, it's normal for a few new hairs to pop up from time to time as you grow older. This happens to genetic women as well. Depending on how many hairs and how much they bother you, you can tweeze them or hunt them down more permanently.
I think the current best advice (and I believe I'm paraphrasing info elsewhere on the forums) is to do several months of laser treatments, then switch to electrolysis once laser has killed the majority of the hairs it can. Both require a skilled practitioner. How effective laser can be is actually a question best asked of a dermatologist. (They have charts!) Electrolysis will be as good as your electrologist.