I'm going to differ here. I'm usually one that lugs my camera everywhere, but on my last trip to Disney, I ditched it and couldn't have made a better decision. I only used my DSLR at Chef Mickey's, but with other family members catching the same pic with their phones, most on my camera didn't get used in the photo book or anything. I had Memory Maker and was VERY pleased by the majority of photos. We even used one of the photopass photog's pictures as our gender reveal announcement in front of Cinderella's castle! (We got the call while at MK that our #2 was a DS!) I say I couldn't have made a better decision because it forced me to be IN the moment instead of recording the moment. I cried as DD saw her first parade because I was watching her, not looking at the screen on my camera. Sure, I videoed her watching the parade the 2nd time with my iphone and the two of us still watch it regularly over a year later.
I think the 2nd benefit of not having the camera there is not WORRYING about the camera being there. I am hesitant about dropping it or leaving it on the stroller or having it around my neck all day. It really is freeing being without it. And honestly, iphones have come such a long way that unless you're planning to do a large, bigger than 8x10 photo, the quality is just fine. Think about it: an iphone camera is much better quality than the point and shoot cameras we were using 10 years ago! However, I would back up your pictures every night from your phone. That way, if it fills up, you can delete them knowing you've backed them up already.