If you don’t get a haircut, your hair will continue to grow longer and longer over time (let’s leave going bald out of it). If you’re a male and don’t shave, your beard will continue to grow longer and longer.
But if you don’t shave the hair on your arms, legs or pubic area, it just seems to reach a limit, and not get any longer.
What is the mechanism by which the body regulates hair growth on its different areas?
Body hair falls out after it gets too long. (Your head hair does, too, but the length limit is a lot lengthier.)
Each hair follicle has an on/off switch. The hair on your head and in pubic regions stays “on” longer.
But that doesn’t answer the OP’s question. The OP is looking for something a bit more specific than “each hair has a different growth cycle.”
That’s kinda obvious.
I can’t believe “lengthier” is actually a word.
Course it is.
Relates to lengthiness.
Whenever asked this question by young children, I’ve always explained it as being the difference between hair and fur. Animals have fur, and humans have fur on their body, but hair on their heads. Fur is smart enough to know when to stop growing, when to thicken up for the winter, and when to shed in the summer, but hair is just too stupid to know all that. That also helps explain the old expression, “dumber that a sack of hair”.
Why the various follicles are “programmed” for their particular hair length is something that has been debated lengthily 
for ages. People have tried to back-construct why humans have so little body hair, pubic underbrush and armpit hair, and longer head hair, even longer and thicker on women than men. One theory held that humans spent a lot of time in the water and children would hitch a ride on mom by her hair, so it had better be long and thick :rolleyes:. Other theories claim that armpits and pubic areas grow more hair to retain secretions that give off odors that act as signals to other humans.
But evolution didn’t leave us the instruction manual and we can never know for sure if any of this is just a result of random mutations that had no pressure to be eliminated, or actually provided some benefit.
Back when I was in college studying biological anthropology, the prevailing theory was that lack of body hair was an adaption to the hot, dry environment in which humans evolved. Body hair became minimal to increase skin exposure to the air, which, combined with sweating, makes an efficient cooling system. Hair is retained on the head because, being on the top of the body, it has the most exposure to the sun and requires some covering to prevent overheating.
As for body hair in the armpits and groin, the only explanation I’ve heard of is that the hair retains pheromones. However, recall that some populations of humans, such as many native North and South American people, do not have body hair.
The OP asked for the mechanism. The article identifies the mechanism. It’s not really that obvious…most people believe that hair reaches a certain length and quits growing.