Accustical or Popcorn texturing ceilings are really tiny bits of perlite mixed with compound that is blown up on the ceiling by a hopper and air compressor. Then, the step that many people ommitt is the painting of the ceiling after this is done. The painting actually glues or adheres the popcorn to the ceiling and makes future painting easier because the stuff does not all fall down. Painting is much easier if sprayed. You can roll it with heavy knap rollers, but it is very time consuming and messy.
Now popcorn ceilings do collect allot of dust and especially around ceiling mounted AC vents. In my opinion these ceilings are a pain in the butt. Many posters on this site are constantly asking how to remove the stuff.
Now diluted down drywall mudd makes for a nice texture if not to "peaky". YOu have your traditional crows foot design and knock down and others to say the least. This does not have all the problems as with textured or accoustical ceilings.
They are easy to roll out when painting and they do not fall down.
The cons of drywall compound texturing is that mostly this is done by professionals that know the pattern and the mixing consistency of the mudd to have moderate peaks. In both applications the room will need to be stripped of all stuff and both are a good days work.
Not sure I answered your question as you wished, but if not, come back with specifics and we will address them individually.