I've dealt with Sand, DE and Cartridge. Sand is the cheapest and easiest to deal with, by far. Matter of fact, I just replaced my 8 year old DE filter with a new sand filter and had zeolite put in, instead of sand. Fantastic job, equal clarity as my DE filter did at its best!
Originally, with sand filters using regular sand, the water wasn't as clear as you could get with a DE filter. No longer true with the new zeolite medium. I only recommend sand filters if you use zeolite, though - otherwise, I prefer DE over cartridge. Zeolite is more expensive than sand, but not prohibitive. This is still cheaper than DE or Cartridge.
DE filters are high maintenance and costly to fix especially if you are not a do it yourselfer. You need to regularly replenish the DE powder.
I found that cartridge filters were a pain in the neck cleaning the cartridges and if you need to replace the cartridges they are expensive.