We have stayed at both, first at GCH (pre-re-model, 2 bedroom suite, Club Level), then DLH (1 bedroom suite) and after one time of that, we have been back to the GCH (post-remodel, Woods/Garden room) several times thereafter. I wholeheartedly prefer GCH over DLH after staying at both.
I didn't think DLH would feel like that much further of a walk than GCH but I was WRONG. There are posters on this board that don't feel it is too far, and they have timed it and measured it, but to me, the feeling of walking just that extra distance was so painful for us (and we exercise regularly and are not at all opposed to walking!) after walking all day in the parks. You actually walk so much more in DLR than WDW, I think, because you park hop by walking (and we hop several times per day). There is no "sitting on the bus" time at DLR, thank goodness, but it makes the feet tired!
I like the style of the GCH much better, with the insanely grand lobby, the stained glass doors, all the Craftsman touches. I felt that the DLH was much more of a generic hotel (I know many here will disagree), and the light up headboards at DLH did not do much for me or the kids for some reason. The pools at DLH are nice, but then so are those at GCH (and at least GCH pools are currently open!). Neither has big slides that appeal to older kids, so no difference there for the ages our kids are now. I did love sitting at Trader Sam's while the kids were in the pools below (they are 11 and 13) at DLH.
The access to DCA is probably the biggest advantage to GCH. It is unbelievably easy to go back and forth. And the security lines to get in are amazingly short any time other than first thing in the morning (and by that I mean they clear within a few minutes of park opening). Get wet on GRR? Go change. Someone having a meltdown? Take one kid back for a break. Need a swim break? Don't we all? Go for it (and book FPs via MP while you sit at the pool for later that night). Cool in the morning but want to ditch the jackets as it warms up? Yep. Grandma worn out and everyone is still going strong? You bet.
The rooms after the refurb are so thoughtfully done, so much nicer than DLH in my opinion. We love the King/sleeper chair/day bed because we get a king and the kids each have their own sleep space, and we don't have to pay for a suite. Balconies!!! And you can see the monorail from many rooms, if you like that. There are great bar options in the evening if your kids are old enough to stay in the room for an hour (thankfully ours are at this point, but wasn't always that way of course).
Overall I should tell you how much I hate GCH so there will be less competition to stay there, but I just love it and so I will share that love here!!