Moment definitely gives you more control over how your final shot turns out — but that also requires more work on your end. The ability to shoot in RAW could be a deal changer for photographers who also shoot on their iPhone, but want the same image-editing capabilities that come with shooting on RAW on a DSLR.
Moment's exposure settings, like shutter speed and ISO, allow you to get more creative with your shots — you can set the shutter speed slower to make moving objects blur, or set it fast to capture objects in motion. If you don't want to individually set exposure or focus settings, you can simply use the autofocus and auto exposure, and take advantage of the RAW shooting capabilities.
Apple's camera is more casual — but also has some features that aren't present in Moment's camera. Portrait mode is missing in Moment, which is a big selling point for the newer iPhones' dual-lens system. In addition, Apple's camera allows for live photos, which isn't available in Moment. The ability to shoot in filters might be important to some users, but filters can easily be applied to photos taken on Moment after they're shot, and Moment gives you more control over the editing process anyway.
So if you want more control over your shots, and the ability to edit in RAW, Moment will probably get the job done for you. But if you just want a solid camera app that allows you to use portrait mode and doesn't require a lot of user input, Apple's native camera app will do just fine. Moment is designed for people who like having the ability to control the minute details of their shots, but that's definitely not everyone.
The decision comes down to what you want out of your phone's camera — more control, or ease of use?