To me being organized is about everything having a place. It is knowing where you are supposed to be and when, and not rushing around stressed out in a panic.
Neatness is different than organization. Neatness is when everything has a home (like my definition of organization), but things are also very clean~ spotless clean. Neatness also really doesn't involve clutter!
You could walk into someones house and think, "wow, this person is so organized!" The reality might be they have the appearance of being clean, but closets, drawers, etc. are a mess. They can't find anything when they need it. Their clutter is all hidden!
I like both. I want to be organized and I want neatness. I believe in order to make this work, we have to start with organizing and once you get in a routine, the neatness piece will become easier.