Some people PM about why it is that important. I will add to this later, but in short, the earlier you apply, the more likely you are to get an interview, the more likely you are to get in. Early on, adcoms and interviewers aren't comparing you to every other applicant, only to a very small number of them. If they want to interview you, they will do it. If they are not sure, they look at your file again next meeting. This means you have the greatest chances of getting an interview early on. Later in the cycle, you are going up against virtually every applicant, almost pushing the system into non-rolling. Not only are you competing with very qualified people, but truthfully, everyone starts looking the same to them. The numbers, research, volunteering still look great, but they have seen it many many times, and even though it may be amazing, at this point they are tryng to add to the diversity of the class; they don't want someone who they've already got. This means not only is it tougher to get an interview, but much more variable as well. If you were to get accepted after applying late, then you will sit there wondering if you applied early, could you have gotten a scholarship?
To see the difference, take a look at UCSF applicants who got an interview on mdapplicant.com. While UCSF obviously doesn't speak for all medical schools, you have to believe many rolling schools have the same basic system (look up any school you would like). Compare the resume of the ones who submitted their primary in June and the ones who did in August/September.
