I'm not an expert, but from what I've read here on the forum, they each have their pros and cons. It's a matter of balancing them out for your own needs.
If I recall, correctly...
Cartridges provide the cleanest filtering, but it is more work to clean.
DE is next best in filtering, but you have to backwash more often, and cleaning and maintenence can be a nuisance.
Sand is lowest on maintenance, doesn't require as much backwashing, is relatively inexpensive, but doesn't filter as tightly as the other two.
There's also a sand substitute called zeolite (aka Zeobrite) which claims to filter down to 3 microns (sand is 30 microns - DE is 2 microns). Some folks, here, love their zeolite, others claim there's not much difference.
Cost is another factor -- both in initial purchase and in operations, but I can't tell you the specifics on that.
I do believe the cartridges can be pricey, but as for cleaning....we just hose ours off about once a month or so. I've heard of people who soak them in various solutions....that's probably more of an issue if you have developed an algae bloom or something. We have never soaked ours in anything. 7th pool season, 2nd filter cartridge. Filter runs 24/7.