I posted this somewhere else, too, but:
I do think applying early gives you a small advantage. But, it's nowhere near as much as people claim here on the board.
These are people who have normally spent years looking at apps, and whether they've seen 1000 over the last month or not, they've gotten to the point where what they've seen before doesn't really matter. They already know what they want to accept.
Now a new professor on the adcomm will probably be swayed a bit more since they haven't had the years of experience. And also, applying early could give you a little boost as a go-get-er who doesn't procrastinate.
I applied the day before the deadline to everywhere but Yale and UT. And Yale was my only rejection. I think I can say that the process was kind to me as my numbers also aren't exactly stellar (though I do have some nice resume experiences).