It's like this:
If you're an All-Star Applicant (I'm talking the whole nine yards, high GPA, high MCAT, interesting life story, great personality, strong LORs, multiple leadership positions, great service activities, solid research experience, being URM or disadvantaged helps too) then you are going to be picked up by one of the top schools that do non-rolling admissions regardless of when you submit.
For those of us who have some issues with our applications, however small or large, it is imperative that we submit early. The same thing goes for people who are "cookie-cutter" applicants. If you have a good application but it's pretty run of the mill then you need to be in early to beat out the other applicants who have almost identical stats/experiences to you. If there is nothing to separate you from the other applicants than you will simply be passed over because they already have a hundred of you interviewing. Sounds harsh, but it's true.
It's a case by case and school by school basis.