While the back office team may check applications for completeness as they are received, adcom won't actually start reviewing applications until after the deadline. (other schools may have different policies)
While submitting a day or two before the deadline is probably prudent to avoid the stress any last minute glitches (eg. power outages, server traffic, hard disk crashes, credit card issues), there is no advantage to your chances in applying earlier within any round. There is however one significant disadvantage - the possibility that you might come up with a better way of expressing things between when you submit the application and the deadline.
Over 50% of applicants submit within 24 hours of the deadline.
FF