Again, trying to not start a "filter war"...
DE will cost more money up-front.
DE filters finer... on a well kept pool, you may or may not see a difference between DE and cartridge.
On restoring a poorly kept pool - DE will get it clearer, faster (but may need a few backwashes).
DE filters, if needed, can be back-washed mid-season if the pressure rises too much (you let the pool go and get algae, or if you live in an area with lots of dust (farm-land)).
Once you are familiar with (key phrase here) your DE filter, you can clean it almost as fast as a cartridge. While you're getting familiar with cleaning it, putting it back together is sometimes a frustrating puzzle.
Always purchase the backwash valve if you go DE.
Cartridge filters run about $100 less than DE.
Cartridges work great on a well kept pool.
On restoring a poorly kept pool - Cartridges will take a lot longer, and may need to be pulled and cleaned several times.
Cartridges will need to be pulled and cleaned mid season if the pressure rises too much (you let the pool go and get algae, or if you live in an area with lots of dust (farm-land)).
Tear down & re-assembly is very straight forward for a cartridge filter.
You cannot backwash a Cartridge filter.
I would never recommend a sand filter. Just a personal bias.