So I have not stayed at Four Seasons. I have stayed at Aulani multiple times, as well as Hilton Hawaiian Waikiki.
I do like Aulani best for several reasons. Everyone in Hawaii is super nice, but I definitely find the staff at Aulani is nicer than any place else. I also find that they are friendlier and more laid back which I appreciate. Even though the Hilton is Waikiki is a gorgeous resort, there's definitely more of a laid back feel at Aulani.
Aulani doesn't feel like "Disney" per se. It has the level of hospitality and attention-to-detail you'd find at a WDW Deluxe resort, but you're not going to have Disney movies and such shoved in your face. You'll see some Disney characters here and there, you can get some merchandise, and you can get Mickey ears on your shave ice. Oh, and your MBs work, which I think is a high point personally.
The big selling point of Aulani is the pool area. The pool at Four Seasons is, well, a pool (which is also how I feel about the Hilton). The pool area at Aulani is spectacular. There's the two waterslides, including the waterslide that drops you into the lazy river. There's an infinity pool. There's also an adults only, two level infinity hot tub overlooking the ocean. It's spectacular, relaxing, I highly recommend it.
The beach is shared. The Four Seasons side does tend to be a little less crowded, but if you go towards that side of the beach where the Aulani chairs are, it's a bit less crowded. You can do all the same excursions at Disney that you do at Four Seasons; they use the same company. If you do choose to stay at Four Seasons, there is a stand run by Disney on the beach where you can grab a Dole Whip, and they do not care if you're a Four Seasons guest.
For the dining options:
Sit down restaurants: I do remember the sit down restaurant at Four Seasons being significantly more expensive than Aulani's options, and Aulani is expensive. You obviously do not have to stay at either of their hotels to use their restaurants. Across the street, you have Monkey Pod, which I recommend. You also have Eggs 'n Things for breakfast. There's several locations around the island; I highly recommend.
Aulani QS:
As in WDW, you can get refillable mugs. They're well worth the money here IMO, and there's Coca Cola freestyle machines, meaning you get even more variety. There's also water stations by the pool.
Aulani has tons of dining options. Ulu café is my go-to. They have the most variety, and their menu rotates. Here's the description from the app.
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You also have the Wailana pool bar, which is my other go-to for lunch.
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Off the Hook was where we'd go if we didn't feel like leaving for dinner.
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There's also another small location that serves coconut shrimp daily, another place that serves shave ice, and a shack out by the beach where you can get Dole Whip. I got watermelon, which I've never found at WDW.
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I do recommend renting a car for at least part of the time, so for days when you're not at the resort, I recommend going elsewhere on the island for dinner. Ko'olina is laid back and far away from the hustle and bustle of Waikiki...but you definitely have fewer dining options.