I've only stayed at the Moana Surfrider (twice), so yes, I did like it. But next time we go back, we'll be staying at the Sheraton. A lot of what we enjoy about the Moana Surfrider can be enjoyed even if you are not staying there. They have a fantastic beach side bar, with live music in the evening, which you would have room charge availability if you stay at the Sheraton. The beach is also very nice right there. Also very easily walked to from the Sheraton.
The Sheraton has some really great pools, Moana Surfrider's pool is rather lacking. Moana Surfrider has never ending wedding photos in their lobby, and you are constantly having to stop as you walk through to avoid ruining someones photos. The historic wing at the Moana has been renovated, but the rooms are REALLY small. Really only fits a king sized bed and the bathroom can accomodate 1 person that can barely turn around.
It all depends on what you want out of where you are staying. They're both close to shopping and dining.
And regardless of where you stay, try the breakfast buffet at Duke's in the Outrigger Waikiki Hotel. Nothing beats breakfast with the ocean view. I think it's about $14ish, compared to the breakfast buffet at the Moana which was $35ish. Enjoy!