This question comes up every year. The answers are generally that if you are a superstar, applying late won't matter. Applying early may make it marginally easier to get an acceptance, because it presumes that not many offers have been made at the time of your application-- from what I can tell there is no data on this and it is speculation.
My annecdotal advice is to just hurry up and submit your applications. Inevitably you (ie applicants) will think that their essays and other such things are way more important and heavily reviewed than they actually are. Just write something good, have it proofread, and submit your app. Do not spend weeks dwelling over these things. While I cannot say for sure that applying early will statistically improve your chances at getting admitted, I do think that it may make your chances of getting an interview easier. And at that point your essays really don't matter much anyway. But just my 2cents.