If the school you are applying to doesn't do rolling admissions, then it does not matter when you get in your application, as long as you get it in by the deadline.
If the school does rolling admissions though, like the previous posters said, applying early is good because the school will interview as they go, so you are competing for more spots (and usually less people) than later on.
I noticed that you said July vs September in your post. Applying in early September is still considered applying "early", the application opens June 30th or July 1st and few people complete their application in July. So usually what people mean by applying early is August to early September
Long story short, take your time to really prepare for the GRE so that you only have to take it one time. As soon as the essay prompt is released, start on working on that, and contact the schools you're interested in to find out if they do rolling admissions and focus on those first. For instance, if those schools require additional essays, write those first. You don't have to apply to all schools at once: you can e-submit for rolling admissions schools, then continue on working on supplemental material for other schools and add them later on. That's what I did
(note that you won't be able to add more schools until your app is verified by PTCAS though,so I wouldn't recommend doing that if you're applying late as the later you apply, the longer it takes PTCAS to verify your app)
Good luck !!