I just finished building a pool 5 months ago with a DE filter and backwashed it for the first time this weekend. About 10 years ago, I had a pool with a cartridge filter, so I'll give you my thoughts on the differences between the two.
Some background....
We are in Arizona with very hard water. I'v heard from many people in AZ to drain your pool every few years to "reset" the water. I am not sure I agree with that yet. I think that's mostly from people who use pucks and their CYA gets out of hand and they don't know how to manage their chemicals correctly. However, I may have to if my calcium levels get too high (I think I am at 600 right now)
My pool with DE is salt water, the last pool with cartridge was pucks.
I had no idea what I was doing 10 years ago with taking care of a pool.
We went with DE for the current pool because I didn't really enjoy the cartridge filter on the last pool. I found cleaning the cartridges a very messy, wet, and time consuming chore (took a few hours of dissasembly, spraying, more spraying, reassembly). However, that might have been because I didn't know what I was doing or I was too OCD and trying to get the cartridges cleaner than necessary. Anyhow, I didn't like it and didn't want to be "constantly" (at least in my mind) cleaning cartridges or replacing them. I was also told that I should be repleneshing water by backwashing anyway, so that my TDS wouldn't get so high. I thought, if I still have to backwash to lower my TDS, then why not just get a filter that you backwash anyway. Enter the DE filter for this pool.
When I decided on DE, I didn't realize you had to replenish the DE when backwashing. That was something I learned a littler later. I was not happy about that and thought it would be an annoyance. Turns out it's cheap and easy to do. I hesitantly backwashed this weekend expecting it to be difficult and take some time. I was pleasantly surprised at how easy it was. The only issues I had were running out of DE (I needed 6 lbs and the PB left me 5) and finding a good place to run the backwash hose to. I ran up to the store and bought DE ($10 for a 10lb bag) and will have some for next time, so problem solved there. For the backwash hose, I am thinking of running some PVC to my sewer cleanout, but that might be more hassle than it's worth. At least next time I will know how fast that water comes rushing out. Adding the DE was pretty easy as well, just mixed it in a bucket with some water and slowly poured it in to the skimmer. Probably took me an hour at most from start to finish.
As others have said, we liked the DE because of how well it can filter (in theory). My wife isn't a fan of bacteria, viruses, and the like and hearing that DE can filter and even possibly "kill" some of that was a great selling point. Plus, knowing that I would probably need to "backwash" (partial drain and refill) for any filter I got, the DE made even more sense. Also, it looks like I will only need to backwash a few times a year, not every few weeks. I am also thinking that cleaning the grids won't be any worse than the cartridge filter and I will have to do it much less. As for losing salt, I am actually having a problem where my salt levels are going up (I am not adding salt, might be in my tap water already) so it's actually good to lose some of that water. Even then, salt is cheap. For the CYA, not too concerned about that, as it's not that much that you'll lose to backwashing and it isn't that expensive to add back.
In the end, I think you would be happy with either type of filter. You just get used to whichever one you have and deal with the pros and cons.
For me, it really was about the PITA of cleaning the cartridge filter every time the pressure gets high. There's no ifs, ands or buts about it, you have to do it. With DE, you only have to clean them occasionally, the rest is just backwashing, adding DE and being done with it.