We have rented cabanas at both, and have loved both experiences.
If I had to choose just one day for a cabana rental, I would choose Aquatica over DC. Aquatica is going to feel MUCH busier than DC in June, as DC caps its admission at a pretty small number while Aquatica does not. For that reason, it will be harder to get a private spot to relax at AQ while it will be less difficult at DC. There are tons of loungers at both parks for relaxing, but those on the Aquatica beaches are quite close together and lined row upon row.
Aquatica's cabana rental, assuming you get a standard, non-pool view one, should be cheaper than DC's (which run at about $175 all the time...AQ has seasonal rates which fluctuate depending on the time of year). Your drinks are already included at DC, where they are not at AQ, so that's an advantage and cost savings too (12 bottles of Dasani water that you won't have to buy at Aquatica). The AQ cabana also provides you with towel rental and a private locker (which are also free at DC, but would be an extra charge at AQ). Plus, the Aquatica cabanas have sides that you can close up and be private on all 4 sides (DC's don't).
DC has some high points too, by providing you with a personal attendant who brings snacks and additional drinks (including your alcohol of choice) to your cabana, while Aquatica does not.
Can't go wrong at either, but if I had to choose.....it would be Aquatica.
