As to comparing sand vs cartridge, I cannot help. Since I've had the pool for just over 18 months I've never even had cloudy water so I can't speak to how long it takes to clear anything up.
So far I need to clean it maybe every 2 or 3 months with very light usage of the pool, just two adults, no kids or dogs. I get a lot of sand in the pool from the waterfall and during pollen season I have more stuff that gets past the skimmer socks. I use the skimmer socks all the time now, that first fall was enough work with cleaning out the filter, much better to keep it in the skimmer where I can get it easier.
Cleaning is not a big deal, remove the band, remove the bottom plug, unseat the air release at the top and when the water stops gushing, open it up and take out a few parts then haul the big carts to the driveway to wash with the brass nozzle thing on the hose. I have used Filter Magic spray but can't find it now and these carts are too big to try to soak so this time I skipped that. Then replace it all, being careful to wash off and lube the O-ring, then "tap and tighten" the band. It does take two of us to get the O-ring on but otherwise it is a one person job.
No idea if washing the carts is more water efficient than backwashing as to gallons used. My carts cost $85 and there are 4. I don't know how old these are so they may be in need of replacement soon as the pool is 10 years old. I have been told that cartridges might last 5 to 7 years, so either way that time is coming up soon.